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Largest product and value

Two positive numbers with a fixed sum have the largest product when they are equal. You write one number as $x$ and the other as the sum minus $x$, and differentiate the product. The same idea applies to a function on a closed interval: check both where the derivative is zero and at the endpoints. The largest of these numbers is the maximum, and the smallest is the minimum.

Symbols

Example

The sum $12$ gives $P(x) = x(12 - x)$ and $x = 6$. The product is $36$. On $[0, 10]$, $f(x) = -x^2 + 6x$ has maximum $f(3) = 9$.

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Part of Mathematics R1: Optimisation.