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.
  • 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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