Three claims about critical points and extrema are listed below.
A. Every local extremum of a differentiable function occurs at a point where the derivative equals zero.
B. Every point where the derivative equals zero corresponds to a local extremum.
C. A function can have an absolute maximum that is not a local maximum.
Respond to each claim.
a. For Claim A, state whether it is always true. Provide a concise justification based on definitions.
b. For Claim B, state whether it is always true. If it is false, construct an explicit example of a differentiable function that illustrates the failure and explain thoroughly.
c. For Claim C, state whether it is always true. If it is true, give an explicit example and explain; if it is false, justify your answer.