Chapter 4.2, Problem 14E

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.f(x) = 1/x, [1, 3]

To determine

To verify: Whether the function f(x)=1x on interval [1,3] satisfies the conditions of Mean Value Theorem and to find all numbers c of the function f(x)=1x on interval [1,3] that satisfy the conclusion of Mean Value Theorem.

Explanation

Given:

The function is f(x)=1x .

Mean Value Theorem:

“Let f be a function that satisfies the following hypothesis:

1. f is continuous on the closed interval [a,b] .

2. f is differentiable on the open interval (a,b) .

Then, there is a number c in (a,b) such that f(c)=f(b)f(a)ba .

Or, equivalently, f(b)f(a)=f(c)(ba) ”.

Verification:

1. The function f(x)=1x is continuous everywhere except the point x=0 .

But notice that 0 does not belongs to the closed interval [1,3] . Therefore, the function f(x) is continuous on the closed interval [1,3] .

2. The function f(x)=1x is differentiable everywhere except at the point x=0 . Therefore, the function f(x) is differentiable on the open interval (1,3) .

Hence, the function f(x) satisfies the conditions of Mean Value Theorem.

Since the number c satisfies the conditions of Mean Value Theorem, it should lie on the open interval (1,3) .

Find the derivative of f(x) .

f(x)=ddx(1x)=1x2

Replace x by c in f(x)

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