Classifying Critical Points and Extreme Values for a Function

Key Questions

• Here is how to find and classify a critical point of $f$.

Remember that $x = c$ is called a critical value of $f$ if $f ' \left(c\right) = 0$ or $f ' \left(c\right)$ is undefined.

$f ' \left(x\right) = 3 {x}^{2} = 0 R i g h t a r r o w x = 0$ is a critical number.

(Note: $f '$ is defined everywhere, $0$ is the only critical value.)

Observing that $f ' \left(x\right) = 3 {x}^{2} \ge 0$ for all $x$,

$f '$ does not change sign around the critical value $0$.

Hence, $f \left(0\right)$ is neither a local maximum nor a local minimum by First Derivative Test.