Chapter 4 | Applications of Derivatives
379
4.4 | The Mean Value Theorem
Learning Objectives
4.4.1 Explain the meaning of Rolle’s theorem. 4.4.2 Describe the significance of the Mean Value Theorem. 4.4.3 State three important consequences of the Mean Value Theorem.
The Mean Value Theorem is one of the most important theorems in calculus. We look at some of its implications at the end of this section. First, let’s start with a special case of the Mean Value Theorem, called Rolle’s theorem. Rolle’s Theorem Informally, Rolle’s theorem states that if the outputs of a differentiable function f are equal at the endpoints of an interval, then there must be an interior point c where f ′( c ) =0. Figure 4.21 illustrates this theorem.
Figure 4.21 If a differentiable function f satisfies f ( a ) = f ( b ), then its derivative must be zero at some point(s) between a and b .
Theorem 4.4: Rolle’s Theorem Let f be a continuous function over the closed interval [ a , b ] and differentiable over the open interval ( a , b ) such that f ( a ) = f ( b ). There then exists at least one c ∈ ( a , b ) such that f ′( c ) =0.
Proof Let k = f ( a ) = f ( b ). We consider three cases: 1. f ( x ) = k for all x ∈ ( a , b ).
2. There exists x ∈ ( a , b ) such that f ( x ) > k . 3. There exists x ∈ ( a , b ) such that f ( x ) < k . Case 1: If f ( x ) = k for all x ∈ ( a , b ), then f ′( x ) =0 for all x ∈ ( a , b ).
Case 2: Since f is a continuous function over the closed, bounded interval [ a , b ], by the extreme value theorem, it has an absolute maximum. Also, since there is a point x ∈ ( a , b ) such that f ( x ) > k , the absolute maximum is greater than k . Therefore, the absolute maximum does not occur at either endpoint. As a result, the absolute maximum must occur at an interior point c ∈ ( a , b ). Because f has a maximum at an interior point c , and f is differentiable at c , by Fermat’s theorem, f ′( c ) =0.
Made with FlippingBook - professional solution for displaying marketing and sales documents online