# Optimization

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**Optimization**, in pure mathematics, is the mathematical analysis of a problem with the goal of finding the most efficient solution. This usually involves maximizing or minimizing a cost 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*,

- , where
*P*is a constant

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.

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.