Difference between revisions of "Optimization"

(Expanded the scope of optimization to include more than just quadratics.)
m (Added the use of Olympiads (AM-GM, Cauchy, etc.).)
 
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** If the quadratic is in the form <math>ax^2 + bx + c</math> (standard form), the maximum or minimum of the quadratic is achieved when <math>x = -\tfrac{b}{2a}</math>.  This can be derived by [[completing the square]].
 
** If the quadratic is in the form <math>ax^2 + bx + c</math> (standard form), the maximum or minimum of the quadratic is achieved when <math>x = -\tfrac{b}{2a}</math>.  This can be derived by [[completing the square]].
 
* The maximum of <math>\sin (x)</math> and <math>\cos (x)</math> is 1, and the minimum of <math>\sin (x)</math> and <math>\cos (x)</math> is -1.
 
* The maximum of <math>\sin (x)</math> and <math>\cos (x)</math> is 1, and the minimum of <math>\sin (x)</math> and <math>\cos (x)</math> is -1.
* One can also use [[coordinate geometry]] to determine the maximum or minimum.  Optimization is often done when two figures touch each other exactly once.
+
* We can use [[inequalities]] like [[AM-GM Inequality]] for some optimization problems.
 +
* We can also use [[coordinate geometry]] to determine the maximum or minimum for some problems.  Optimization is often done when two figures touch each other exactly once.
 
* In [[calculus]], for a function <math>f(x)</math>, the local maximums and local minimums are part of the critical points of the function.  The x-values of the critical points can be found by taking the derivative of <math>f(x)</math> and setting it to equal 0.  In order to find the absolute maximum or minimum, one needs to also check the endpoints of an interval.
 
* In [[calculus]], for a function <math>f(x)</math>, the local maximums and local minimums are part of the critical points of the function.  The x-values of the critical points can be found by taking the derivative of <math>f(x)</math> and setting it to equal 0.  In order to find the absolute maximum or minimum, one needs to also check the endpoints of an interval.
  
 
[[Category: Algebra]]
 
[[Category: Algebra]]
 
[[Category: Calculus]]
 
[[Category: Calculus]]

Latest revision as of 14:11, 14 April 2020

Optimization is simply finding the maximum or minimum possible value. In order to prove that a value is a maximum or minimum, one needs to prove that the value is attainable and that there is no higher or lower value (depending on the problem) that works.

Optimization Techniques

  • There are multiple ways to determine the maximum or minimum (depending of the leading term) of a quadratic (depending on the form).
    • If the quadratic is in the form $a(x-h)^2+k$ (vertex form), the maximum or minimum of the quadratic is $k$ by the Trivial Inequality.
    • If the quadratic is in the form $ax^2 + bx + c$ (standard form), the maximum or minimum of the quadratic is achieved when $x = -\tfrac{b}{2a}$. This can be derived by completing the square.
  • The maximum of $\sin (x)$ and $\cos (x)$ is 1, and the minimum of $\sin (x)$ and $\cos (x)$ is -1.
  • We can use inequalities like AM-GM Inequality for some optimization problems.
  • We can also use coordinate geometry to determine the maximum or minimum for some problems. Optimization is often done when two figures touch each other exactly once.
  • In calculus, for a function $f(x)$, the local maximums and local minimums are part of the critical points of the function. The x-values of the critical points can be found by taking the derivative of $f(x)$ and setting it to equal 0. In order to find the absolute maximum or minimum, one needs to also check the endpoints of an interval.
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