Chapter 4.2, Problem 12E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Using Rolle's Theorem In Exercises 11-26, determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of e in the open interval (a, b) such that f ' ( x ) = 0 . If Rolle's Theorem cannot be applied, explain why not. f ( x ) = x 2 − 8 x + 5 , [ 2 , 6 ]

To determine
Whether Rolle’s Theorem can be applied to f(x)=x28x+5 on the closed interval [2,6] and if yes, then the values of c in the open interval (2,6) such that f(c)=0.

Explanation

Since, f(x)=x2âˆ’8x+5 is a quadratic function, therefore it is defined for all values of x and is continuous on the closed interval [2,6].

Now, take derivative of the function f(x)=x2âˆ’8x+5 as,

fâ€²(x)=2xâˆ’8

Since, fâ€²(x) is a linear function and is defined for all values of x in the open interval (2,6), f(x)=x2âˆ’8x+5 is differentiable on the open interval (2,6).

Now, for the interval (2,6), a=2Â andÂ b=6 so check for f(a)=f(b),

f(2)=(2)

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