Question

Proof:
Suppose that f is continuous on [a,b] and differentiable on (a,b) and that m<= f'(x) <= M on (a,b). Use the racetrack principle to prove that f(x)-f(a)<=M(x-a) for all x in [a,b], and that m(x-a)<= f(x)-f(a) for all x in [a,b]. Conclude that m<= (f(b)-f(a))/b-a) <= M.