How do you find and classify the critical points of $f(x)=x^3$ ?

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Answer 1 · 2014-09-28T11:43:46

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

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

$f'(x)=3x^2=0 Rightarrow x=0$ is a critical number.

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

Observing that $f'(x)=3x^2 ge 0$ for all $x$ ,

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

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

How do you find and classify the critical points of f(x)=x^3f(x)=x^3?