Optimization

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.
One can also use coordinate geometry to determine the maximum or minimum. 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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Optimization

Optimization Techniques