Unit 5: Analytical Applications of Differentiation

f must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Both must be stated in a justification.

There is at least one c in (a, b) with f'(c) = (f(b) − f(a))/(b − a): some instantaneous rate equals the average rate.

At some interior point the tangent line is parallel to the secant line joining (a, f(a)) and (b, f(b)).

If also f(a) = f(b), then some c in (a, b) has f'(c) = 0 (a horizontal tangent).

f is not differentiable at x = 0, which is inside (−1, 1). Indeed no c gives f'(c) = 0, the average rate.

f is not continuous on [−1, 1]: it is undefined at x = 0. The average rate is 1, yet f'(x) = −1/x^2 is always negative.