Difference between revisions of "Optimization"
Rockmanex3 (talk | contribs) (Expanded the scope of optimization to include more than just quadratics.) |
Rockmanex3 (talk | contribs) 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. | ||
| − | * | + | * 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]] | ||
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
(vertex form), the maximum or minimum of the quadratic is 
by the Trivial Inequality. - If the quadratic is in the form
(standard form), the maximum or minimum of the quadratic is achieved when 
. This can be derived by completing the square.
- If the quadratic is in the form
- The maximum of
and 
is 1, and the minimum of and 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
, 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 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.
(vertex form), the maximum or minimum of the quadratic is 
by the 
(standard form), the maximum or minimum of the quadratic is achieved when 
. This can be derived by 
and 
is 1, and the minimum of 
, 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 
