Suppose that f is differentiable on [a, b], but the derivative may be discontinuous. (a) Suppose that f ' (a) 0 f ' (b). Show that the minimum of...

Suppose that f is differentiable on [a, b], but the derivative may be discontinuous.

(a) Suppose that f ' (a) < 0 < f ' (b). Show that the minimum of f does not occur at an endpoint. What can you conclude?

(b) Suppose that f ' (a) < L < f ' (b). Prove that there is an c ∈ (a, b) such that f ' (c) = L.

(c) Prove that if f ' (x) is monotone, then it is continuous.