Schaum's Outline of Theory and Problems of Beginning Calculus

Chapter 17The Mean-Value Theorem andthe Sign of the Derivative17.1 ROLLE’S THEOREM AND THE MEAN-VALUE THEOREMLet us consider a function f that is continuous over a closed interval [a, b] and differentiable atevery point of the open interval (a, b). We also suppose thatf(a) = f (b) = 0. Graphs of some examples ofsuch a function are shown in Fig. 17-1. It seems clear that there must always be some point betweenx = a and x = b at which the tangent line is horizontal and, therefore, at which the derivative offis 0.YYa b X(6)Fig. 17-1Theorem 17.1 (Rolle’s Theorem): Iffis continuous over a closed interval [a, b], differentiable on theopen interval (a, b), and if f(a) = f (b) = 0, then there is at least one number c in (a, b)such thatf’(c) = 0.See Problem 17.6 for the proof.Rolle’s theorem enables us to prove the following basic theorem (which is also referred to as the lawof the mean for derivatives).Theorem 17.2 (Mean-Value Theorem): Letfbe continuous over the closed interval [a, b] and differentiableon the open interval (a, b). Then there is a number c in the open interval (a, b) suchthatFor a proof, see Problem 17.7.EXAMPLE In graphic terms, the mean-value theorem states that at some point along an arc of a curve, thetangent line is parallel to the line connecting the initial and the terminal points of the arc. This can be seenin Fig. 17-2, where there are three numbers (c1, c2, and cj) between a and b for which the slope of thetangent line to the graph f ’(c) is equal to the slope of the line AB,129f (b) - f (4b-a *