Question

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

Answer 1

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.