Yet another old one, from the 2010 Specialist Exam 2 (CAS). We’ll look to semi-update with excerpts from the examination report once people have had a chance to comment. (Readers are of course free to peek at the examination report, anytime they so wish.)
The following is the (slightly tidied) solution to (c) in the examination report:
which gives the stated result.
The question is generally aimless and annoying, but the main issue is (c). It makes absolutely no sense to “find the value of a”, since the number a is in no sense defined. Worse, the report’s method to find the maximum value of a is fundamentally invalid. The domain of the derivative of f’ can only be found by considering when the function f can be differentiated, and definitely not by considering the final form of f’.