SL 5.6—Stationary points

A stationary point occurs at an interior value where $f'(x)=0$. Use technology when appropriate to generate $f'(x)$ and solve for the x-values, then calculate the corresponding y-values.

Classify a local maximum when the function changes from increasing to decreasing and a local minimum when it changes from decreasing to increasing. Compare endpoints as well when the greatest or least value on a restricted domain is required.

Example

For $f(x)=x^3-3x$, $f'(x)=3x^2-3=0$ at $x=\pm1$. The derivative changes $+$ to $-$ at $-1$ and $-$ to $+$ at $1$, giving a local maximum and minimum respectively.

$f'(x)=0$ alone does not guarantee an extremum, and a local extremum need not be the absolute extremum on the stated domain. The second-derivative test belongs to AHL 5.10.