A function h is defined on the closed interval [1,5] by
h(x)=⎩⎨⎧x−3x2−9,k,x=3x=3
where k is a real constant.
(1) Part 1: (a) Find the value of k that makes h continuous on [1,5]. Justify your answer.
(2) Part 2: (b) For the value of k obtained in part (a), determine whether h is differentiable at x=3. Support your conclusion with an appropriate limit or derivative computation.
(3) Part 3: (c) Using your work in parts (a) and (b), decide whether the hypotheses of the Mean Value Theorem are satisfied for h on [1,5]. Provide a detailed explanation.
(4) Part 4: (d) If the hypotheses are satisfied, compute every value c in (1,5) that satisfies h′(c) equal to the average rate of change of h over [1,5]. Present complete algebraic work.