**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* = 2*l* + 2*w* 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 2*l* = *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.