Question: 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.