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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 = 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.

{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.

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