Let g(x)=x1/3(x−2).
The function g is defined for all real numbers and is differentiable for all x=0.
Answer the questions that follow.
(1) Determine all critical points of g on the interval [−1,3] and classify each as a local maximum, local minimum, or neither. Give full justification, including any derivative analysis required.
(2) Prove that g attains an absolute maximum value and an absolute minimum value on [−1,3]. State the points at which these global extrema occur and provide complete reasoning.
(3) Explain why the conclusion in part (b) would fail if the interval were changed to (−1,3]. Construct a rigorous argument that includes any necessary counterexample features.