(a) Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if f: (a, b) → R is an unbounded function (i.e., for every M > 0 there exists x € (a, b) such that |f (x)|...

Extracted text: (a) Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if f: (a, b) → R is an unbounded function (i.e., for every M > 0 there exists x € (a, b) such that |f (x)| > M), then f' is also an unbounded function. (b) Give an example of a function f: (a, b) → R, where (a, b) is a bounded open real interval, such that f is differentiable on (a, b), f' is unbounded on (a, b), but f is bounded function on (a, b). 2.