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Convex function - Revision history
2024-03-29T16:00:06Z
Revision history for this page on the wiki
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https://artofproblemsolving.com/wiki/index.php?title=Convex_function&diff=76482&oldid=prev
Data2000 at 04:10, 20 February 2016
2016-02-20T04:10:51Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 04:10, 20 February 2016</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>A <del class="diffchange diffchange-inline">[[</del>function<del class="diffchange diffchange-inline">]] <math>f: I \mapsto \mathbb{R}</math> for some interval <math> \displaystyle I \subseteq \mathbb{R} </math> </del>is <del class="diffchange diffchange-inline">''convex'' (sometimes written ''concave up'') over <math> \displaystyle I </math> if and only if </del>the <del class="diffchange diffchange-inline">set </del>of <del class="diffchange diffchange-inline">all points <math> \displaystyle (x,y) </math> such that <math> \displaystyle y \ge f(x) </math> is [[convex set | convex]].  Equivalently, <math> \displaystyle f </math> is convex if for </del>every <del class="diffchange diffchange-inline"><math> \lambda \</del>in <del class="diffchange diffchange-inline">[0,1] </math> and every <math> x,y \in I</math>,</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>A <ins class="diffchange diffchange-inline">convex </ins>function is <ins class="diffchange diffchange-inline">a continuous function whose value at </ins>the <ins class="diffchange diffchange-inline">midpoint </ins>of every <ins class="diffchange diffchange-inline">interval </ins>in <ins class="diffchange diffchange-inline">its domain does not exceed the arithmetic mean of its values at the ends of the interval</ins>.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><center><math> \displaystyle \lambda f(x) + (1-\lambda)f(y) \ge f\left( \lambda x + (1-\lambda) y \right) </math>.</center></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">We say that <math> \displaystyle f </math> is '''strictly convex''' if equality occurs only when <math> \displaystyle x=y </math> or <math> \lambda \in \{ 0,1 \} </math></del>.</div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Usually</del>, <del class="diffchange diffchange-inline">when we do not specify </del><math> \<del class="diffchange diffchange-inline">displaystyle </del>I </math>, <del class="diffchange diffchange-inline">we mean </del><math> <del class="diffchange diffchange-inline">I </del>= \<del class="diffchange diffchange-inline">mathbb</del>{<del class="diffchange diffchange-inline">R</del>} </math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">A [[function]] <math>f: I \mapsto \mathbb{R}</math> for some interval <math>I \subseteq \mathbb{R} </math> is ''convex'' (sometimes written ''concave up'') over <math>I </math> if and only if the set of all points <math>(x,y) </math> such that <math>y \ge f(x) </math> is [[convex set | convex]].  Equivalently, <math>f </math> is convex if for every <math> \lambda \in [0</ins>,<ins class="diffchange diffchange-inline">1] </math> and every </ins><math> <ins class="diffchange diffchange-inline">x,y </ins>\<ins class="diffchange diffchange-inline">in </ins>I</math>,</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><center><math>\lambda f(x) + (1-\lambda)f(y) \ge f\left( \lambda x + (1-\lambda) y \right) </ins><<ins class="diffchange diffchange-inline">/</ins>math><ins class="diffchange diffchange-inline">.</center></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">We say that <math>f </math> is '''strictly convex''' if equality occurs only when <math>x</ins>=<ins class="diffchange diffchange-inline">y </math> or <math> \lambda \in </ins>\{ <ins class="diffchange diffchange-inline">0,1 \</ins>} </math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">We say that </del><math> <del class="diffchange diffchange-inline">\displaystyle f </del></math> <del class="diffchange diffchange-inline">is (strictly) '''concave''' (or, occasionally</del>, <del class="diffchange diffchange-inline">that it is ''concave down'') if </del><math> \<del class="diffchange diffchange-inline">displaystyle -f </del></math> <del class="diffchange diffchange-inline">is (strictly) convex</del>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Usually, when we do not specify </ins><math><ins class="diffchange diffchange-inline">I </ins></math>, <ins class="diffchange diffchange-inline">we mean </ins><math> <ins class="diffchange diffchange-inline">I = </ins>\<ins class="diffchange diffchange-inline">mathbb{R} </ins></math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">If </del><math> <del class="diffchange diffchange-inline">\displaystyle </del>f </math> is <del class="diffchange diffchange-inline">differentiable on an interval <math> \displaystyle I </math></del>, <del class="diffchange diffchange-inline">then </del>it is <del class="diffchange diffchange-inline">convex on <math> \displaystyle I </math> </del>if <del class="diffchange diffchange-inline">and only if <math> \displaystyle f' </del><<del class="diffchange diffchange-inline">/</del>math> <del class="diffchange diffchange-inline">is non</del>-<del class="diffchange diffchange-inline">decreasing on <math> \displaystyle I </math>.  Similarly, if <math> \displaystyle </del>f </math> is <del class="diffchange diffchange-inline">twice differentiable over an interval <math> \displaystyle I </math>, we say it is convex over <math> \displaystyle I </math> if and only if <math> f''</del>(<del class="diffchange diffchange-inline">x</del>) <del class="diffchange diffchange-inline">\ge 0 </math> for all <math> x \in I </math></del>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">We say that </ins><math>f </math> is <ins class="diffchange diffchange-inline">(strictly) '''concave''' (or, occasionally</ins>, <ins class="diffchange diffchange-inline">that </ins>it is <ins class="diffchange diffchange-inline">''concave down'') </ins>if <math>-f </math> is (<ins class="diffchange diffchange-inline">strictly</ins>) <ins class="diffchange diffchange-inline">convex</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Note that in our previous paragraph, our requirements that <math> <del class="diffchange diffchange-inline">\displaystyle </del>f </math> is differentiable and twice differentiable are crucial.  For a simple example, consider the function</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">If <math>f </math> is differentiable on an interval <math>I </math>, then it is convex on <math>I </math> if and only if <math>f' </math> is non-decreasing on <math>I </math>.  Similarly, if <math>f </math> is twice differentiable over an interval <math>I </math>, we say it is convex over <math>I </math> if and only if <math> f''(x) \ge 0 </math> for all <math> x \in I </math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Note that in our previous paragraph, our requirements that <math>f </math> is differentiable and twice differentiable are crucial.  For a simple example, consider the function</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><center></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></center></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>defined over the non-negative reals.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>defined over the non-negative reals.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>It is piecewise differentiable, but at infinitely many points (for all natural numbers <math> <del class="diffchange diffchange-inline">\displaystyle </del>x </math>, to be exact) it is not differentiable.  Nevertheless, it is convex.  More significantly, consider the function</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>It is piecewise differentiable, but at infinitely many points (for all natural numbers <math>x </math>, to be exact) it is not differentiable.  Nevertheless, it is convex.  More significantly, consider the function</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><center></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></center></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></center></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>over the interval <math> <del class="diffchange diffchange-inline">\displaystyle </del>[-2, 2] </math>.  It is continuous, and twice differentiable at every point except <math> <del class="diffchange diffchange-inline">\displaystyle</del>{} (0, 1) </math>.  Furthermore, its second derivative is greater than 0, wherever it is defined.  But its graph is shaped like a curvy W, and it is not convex over <math> <del class="diffchange diffchange-inline">\displaystyle </del>[-2,2] </math>, although it is convex over <math> <del class="diffchange diffchange-inline">\displaystyle </del>[-2,0] </math> and over <math> <del class="diffchange diffchange-inline">\displaystyle </del>[0,2] </math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>over the interval <math>[-2, 2] </math>.  It is continuous, and twice differentiable at every point except <math>{} (0, 1) </math>.  Furthermore, its second derivative is greater than 0, wherever it is defined.  But its graph is shaped like a curvy W, and it is not convex over <math>[-2,2] </math>, although it is convex over <math>[-2,0] </math> and over <math>[0,2] </math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{stub}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{stub}}</div></td></tr>
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Data2000
https://artofproblemsolving.com/wiki/index.php?title=Convex_function&diff=15227&oldid=prev
Boy Soprano II at 14:17, 17 June 2007
2007-06-17T14:17:47Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 14:17, 17 June 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A [[function]] <math>f: I \mapsto \mathbb{R}</math> for some interval <math> \displaystyle I \subseteq \mathbb{R} </math> is ''convex'' (sometimes written ''concave up'') over <math> \displaystyle I </math> if and only if the set of all points <math> \displaystyle (x,y) </math> such that <math> \displaystyle y \ge f(x) </math> is [[convex set | convex]].  Equivalently, <math> \displaystyle f </math> is convex if for every <math> \lambda \in [0,1] </math> and every <math> x,y \in I</math>,</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A [[function]] <math>f: I \mapsto \mathbb{R}</math> for some interval <math> \displaystyle I \subseteq \mathbb{R} </math> is ''convex'' (sometimes written ''concave up'') over <math> \displaystyle I </math> if and only if the set of all points <math> \displaystyle (x,y) </math> such that <math> \displaystyle y \ge f(x) </math> is [[convex set | convex]].  Equivalently, <math> \displaystyle f </math> is convex if for every <math> \lambda \in [0,1] </math> and every <math> x,y \in I</math>,</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><center><math> \displaystyle \lambda f(x) + (1-\lambda)f(y) \ge f\left( \lambda x + (1-\lambda) y \right) </math>.</center> <del class="diffchange diffchange-inline"> Usually, </del>when <del class="diffchange diffchange-inline">we do not specify </del><math> \displaystyle <del class="diffchange diffchange-inline">I </del></math><del class="diffchange diffchange-inline">, we mean </del><math> <del class="diffchange diffchange-inline">I = </del>\<del class="diffchange diffchange-inline">mathbb</del>{<del class="diffchange diffchange-inline">R</del>} </math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><center><math> \displaystyle \lambda f(x) + (1-\lambda)f(y) \ge f\left( \lambda x + (1-\lambda) y \right) </math>.</center></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">We say that <math> \displaystyle f </math> is '''strictly convex''' if equality occurs only </ins>when <math> \displaystyle <ins class="diffchange diffchange-inline">x=y </ins></math> <ins class="diffchange diffchange-inline">or </ins><math> \<ins class="diffchange diffchange-inline">lambda \in \</ins>{ <ins class="diffchange diffchange-inline">0,1 \</ins>} </math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">We say that </del><math> \displaystyle <del class="diffchange diffchange-inline">f </del></math> <del class="diffchange diffchange-inline">is '''concave''' (or</del>, <del class="diffchange diffchange-inline">occasionally, that it is ''concave down'') if </del><math> \<del class="diffchange diffchange-inline">displaystyle -f </del></math> <del class="diffchange diffchange-inline">is convex</del>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Usually, when we do not specify </ins><math> \displaystyle <ins class="diffchange diffchange-inline">I </ins></math>, <ins class="diffchange diffchange-inline">we mean </ins><math> <ins class="diffchange diffchange-inline">I = </ins>\<ins class="diffchange diffchange-inline">mathbb{R} </ins></math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>If <math> \displaystyle f </math> is differentiable, then it is convex if and only if <math> \displaystyle f' </math> is non-decreasing.  Similarly, if <math> \displaystyle f </math> is twice differentiable, we say it is convex over <del class="diffchange diffchange-inline">an interval </del><math> \displaystyle I </math> if and only if <math> f(x) \ge 0 </math> for all <math> x \in I </math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">We say that <math> \displaystyle f </math> is (strictly) '''concave''' (or, occasionally, that it is ''concave down'') if <math> \displaystyle -f </math> is (strictly) convex.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>If <math> \displaystyle f </math> is differentiable <ins class="diffchange diffchange-inline">on an interval <math> \displaystyle I </math></ins>, then it is convex <ins class="diffchange diffchange-inline">on <math> \displaystyle I </math> </ins>if and only if <math> \displaystyle f' </math> is non-decreasing <ins class="diffchange diffchange-inline">on <math> \displaystyle I </math></ins>.  Similarly, if <math> \displaystyle f </math> is twice differentiable <ins class="diffchange diffchange-inline">over an interval <math> \displaystyle I </math></ins>, we say it is convex over <math> \displaystyle I </math> if and only if <math> f<ins class="diffchange diffchange-inline">''</ins>(x) \ge 0 </math> for all <math> x \in I </math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Note that in our previous paragraph, our requirements that <math> \displaystyle f </math> is differentiable and twice differentiable are crucial.  For a simple example, consider the function</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Note that in our previous paragraph, our requirements that <math> \displaystyle f </math> is differentiable and twice differentiable are crucial.  For a simple example, consider the function</div></td></tr>
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<td colspan="2" class="diff-lineno">Line 25:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{stub}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{stub}}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Resources ==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Jensen's Inequality]]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Karamata's Inequality]]</ins></div></td></tr>
</table>
Boy Soprano II
https://artofproblemsolving.com/wiki/index.php?title=Convex_function&diff=14429&oldid=prev
Boy Soprano II: People! Not all functions are twice differentiable!!!
2007-04-09T02:21:43Z
<p>People! Not all functions are twice differentiable!!!</p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 02:21, 9 April 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Let </del><math>f:\mathbb{R}\<del class="diffchange diffchange-inline">to</del>\mathbb{R}</math> <del class="diffchange diffchange-inline">be a function, </del>and <del class="diffchange diffchange-inline">let </del><math>(<del class="diffchange diffchange-inline">a</del>,<del class="diffchange diffchange-inline">b</del>)</math> <del class="diffchange diffchange-inline">be an </del>[[<del class="diffchange diffchange-inline">open interval</del>]]. <del class="diffchange diffchange-inline">(This can be done with </del>[<del class="diffchange diffchange-inline">[closed interval</del>]<del class="diffchange diffchange-inline">]s or [[half</del>-<del class="diffchange diffchange-inline">open interval]]s as well</del>.<del class="diffchange diffchange-inline">) Then </del><math>f</math> is <del class="diffchange diffchange-inline">said to be </del>'''<del class="diffchange diffchange-inline">convex</del>''' <del class="diffchange diffchange-inline">on </del><math><del class="diffchange diffchange-inline">(a</del>,<del class="diffchange diffchange-inline">b)</del></math> if <math>f<del class="diffchange diffchange-inline">''</del>(x)\ge 0</math> for all <math>x\in(a,<del class="diffchange diffchange-inline">b)</del></math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">A [[function]] </ins><math>f: <ins class="diffchange diffchange-inline">I \mapsto </ins>\mathbb{R}<ins class="diffchange diffchange-inline"></math> for some interval <math> \displaystyle I </ins>\<ins class="diffchange diffchange-inline">subseteq </ins>\mathbb{R} </math> <ins class="diffchange diffchange-inline">is ''convex'' (sometimes written ''concave up'') over <math> \displaystyle I </math> if </ins>and <ins class="diffchange diffchange-inline">only if the set of all points </ins><math> <ins class="diffchange diffchange-inline">\displaystyle </ins>(<ins class="diffchange diffchange-inline">x</ins>,<ins class="diffchange diffchange-inline">y</ins>) </math> <ins class="diffchange diffchange-inline">such that <math> \displaystyle y \ge f(x) </math> is </ins>[[<ins class="diffchange diffchange-inline">convex set | convex</ins>]]. <ins class="diffchange diffchange-inline"> Equivalently, <math> \displaystyle f </math> is convex if for every <math> \lambda \in </ins>[<ins class="diffchange diffchange-inline">0,1</ins>] <ins class="diffchange diffchange-inline"></math> and every <math> x,y \in I</math>,</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><center><math> \displaystyle \lambda f(x) + (1-\lambda)f(y) \ge f\left( \lambda x + (1</ins>-<ins class="diffchange diffchange-inline">\lambda) y \right) </math>.</center>  Usually, when we do not specify <math> \displaystyle I </math>, we mean <math> I = \mathbb{R} </math></ins>.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">We say that </ins><math> <ins class="diffchange diffchange-inline">\displaystyle </ins>f </math> is '''<ins class="diffchange diffchange-inline">concave''' (or, occasionally, that it is '</ins>'<ins class="diffchange diffchange-inline">concave down</ins>''<ins class="diffchange diffchange-inline">) if </ins><math> <ins class="diffchange diffchange-inline">\displaystyle -f </math> is convex.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">If <math> \displaystyle f </math> is differentiable, then it is convex if and only if <math> \displaystyle f' </math> is non-decreasing.  Similarly, if <math> \displaystyle f </math> is twice differentiable</ins>, <ins class="diffchange diffchange-inline">we say it is convex over an interval <math> \displaystyle I </ins></math> <ins class="diffchange diffchange-inline">if and only </ins>if <math> f(x) \ge 0 </math> for all <math> x \in <ins class="diffchange diffchange-inline">I </math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Note that in our previous paragraph, our requirements that <math> \displaystyle f </math> is differentiable and twice differentiable are crucial.  For a simple example, consider the function</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><center></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">f(x) = \lfloor x \rfloor </ins>(<ins class="diffchange diffchange-inline">x - \lfloor x \rfloor ) + {\lfloor x \rfloor \choose 2}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"></math>,</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"></center></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">defined over the non-negative reals.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">It is piecewise differentiable, but at infinitely many points (for all natural numbers <math> \displaystyle x </math>, to be exact) it is not differentiable.  Nevertheless, it is convex.  More significantly, consider the function</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><center></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">f(x) = \left( |x| - 1 \right)^2</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"></math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"></center></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">over the interval <math> \displaystyle [-2, 2] </math>.  It is continuous, and twice differentiable at every point except <math> \displaystyle{} (0, 1) </math>.  Furthermore, its second derivative is greater than 0, wherever it is defined.  But its graph is shaped like </ins>a <ins class="diffchange diffchange-inline">curvy W, and it is not convex over <math> \displaystyle [-2</ins>,<ins class="diffchange diffchange-inline">2] </math>, although it is convex over <math> \displaystyle [-2,0] </math> and over <math> \displaystyle [0,2] </ins></math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{stub}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{stub}}</div></td></tr>
</table>
Boy Soprano II
https://artofproblemsolving.com/wiki/index.php?title=Convex_function&diff=5263&oldid=prev
ComplexZeta at 05:59, 29 June 2006
2006-06-29T05:59:10Z
<p></p>
<p><b>New page</b></p><div>Let <math>f:\mathbb{R}\to\mathbb{R}</math> be a function, and let <math>(a,b)</math> be an [[open interval]]. (This can be done with [[closed interval]]s or [[half-open interval]]s as well.) Then <math>f</math> is said to be '''convex''' on <math>(a,b)</math> if <math>f''(x)\ge 0</math> for all <math>x\in(a,b)</math>.<br />
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{{stub}}</div>
ComplexZeta