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Maxima and minima

A maximum or a minimum lies where the derivative is zero and changes sign. You solve $f'(x) = 0$, insert the value into $f$, and read the sign chart. If the derivative is positive on the left and negative on the right, it is a maximum. The second derivative says the same thing: a negative second derivative means a maximum, and a positive one means a minimum.

Symbols

Example

$f(x) = x^2 - 4x$ has $f'(x) = 2x - 4 = 0$ when $x = 2$. Then $f(2) = 4 - 8 = -4$, a minimum.

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Part of Mathematics R1: Differentiation and curve analysis.