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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,

, 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. It can also be found by completing the square.

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.