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

1 Answer
Sep 28, 2014

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.