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'(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.

Questions