Optimization, in pure mathematics, is the mathematical analysis of a problem with the goal of finding the most efficient solution, or one that maximizes or minimizes a function.

A common example is finding the dimensions of a rectangle such that the area is maximized for a given perimeter. Using the perimeter equation P = 2l + 2w and the area equation A = lw,

$ {w} = \frac{\left(P - 2l\right)}{2} $
$ {A} = \frac{l\left({P - 2l}\right)}2 $, where P is a constant
$ {A} = \frac{lP}{2} - l^2 $
$ {A} = -l^2 + \frac{lP}{2} $

This gives a quadratic equation. The precise value of the optimal length l can be found by taking the derivative of the equation and finding the root. It can also be found by completing the square.

$ {A} = -l^2 + \frac{lP}{2} $
$ \frac{dA}{dl} = -2l + \frac{P}{2} $
$ {0} = -2l + \frac{P}{2} $
$ {2l} = \frac{P}{2} $
$ {l} = \frac{P}{4} $

We can infer from this that the optimal rectangle is a square, as 2l = P/2, which requires w to also be P/4, making the length and width equal. This should also match the intutitive thought, where you would use a square to maximize the area of a rectangle.