Chapter 3.10, Problem 28E

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. This is called the Mean Value Inequality. In words: If the instantaneous rate of change of f is between m and M on an interval, so is the average rate of change of f over the interval.