A function is defined by
f(x)={xsinx,k,x=0x=0
on the closed interval [−π,π], where k is a real constant.
(a) Determine the value of k that makes f continuous on [−π,π]. Give a clear justification based on the definition of continuity.
(b) For the value of k found in part (a), decide whether f is differentiable at x=0. Support your conclusion with an appropriate limit calculation or reasoning.
(c) Using your results from parts (a) and (b), state and justify whether the hypotheses of the Mean Value Theorem are satisfied for f on [−π,π]. Explain your reasoning in complete sentences.