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Chapter 3: Q. 68 (page 249)

Follow the method of proof that we used for Rolle鈥檚 Theorem to prove the following slightly more general theorem: If fis continuous on [a,b]and differentiable on (a,b), and if f(a)=f(b), then there is some value c鈭�(a,b)with f(c)=0.

Short Answer

Expert verified

The given theorem is proved by Rolle's Theorem.

Step by step solution

01

Step 1. Given Information.

fis continuous on [a,b] and differentiable on(a,b).

02

Step 2. Using Rolle's Theorem.

We have to show that iff(a)=f(b), then there is some value c鈭�(a,b)with f'(c)=0.

Let us consider the Extreme Value Theorem if a function f(x)is continuous on the closed interval [a,b]and differentiable on the open interval (a,b). Then there exists at least one value c鈭�(a,b)such that f'(c)=f(b)-f(a)b-aas,

f'(c)=f(b)-f(a)b-a=f(b)-f(b)b-a=0b-a=0

Hence, if f(a)=f(b)then exists c鈭�(a,b)withf'(c)=0.

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