Theorem 5.2.7 (Darboux's Theorem). If f is differentiable on an interval [a, b], and if a satisfies f'(a) < a < f'(b) (or f'(a) > a > f'(b)), then there exists a point c E (a, b) where f'(c) = a. Proof. We first simplify matters by defining a new function g(x) = f(x) — ax on [a, b]. Notice that g is differentiable on [a, b] with g'(x) = f'(x)-a. In terms of g, our hypothesis states that g'(a) < 0 < g'(b), and we hope to show that g'(c) 0 for some c € (a, b). The remainder of the argument is outlined in Exercise 5.2.6. Exercise 5.2.6. (a) Assume that g is differentiable on [a, b] and satisfies g'(a) < 0 g'(b). Show that there exists a point x = (a, b) where g(a) > g(x), and a point y (a, b) where g(y) < g(b). (b) Now complete the proof of Darboux's Theorem started earlier.