   Chapter 4.2, Problem 10E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

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

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Let f(x) = tan x. Show that f(0) = f(π) but there is no number c in (0, π) such that f′(c) = 0. Why does this not contradict Rolle’s Theorem?

To determine

To show: For the function f(x)=tanx , f(0)=f(π) but it has no number c on the open interval (0,π) such that f(c)=0 ; justify that why this not contradict Rolle’s Theorem.

Explanation

Rolle’s Theorem:

“Let a function f satisfies the following conditions

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

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

3. f(a)=f(b)

Then, there is a number c in open interval (a,b) such that f(c)=0 .”

Given:

The function f(x)=tanx .

Proof:

Find the values of f(x) for the end points of the given interval.

Substituting x=0 in f(x) ,

f(0)=tan0=0[tan0=0]

Substituting x=π in f(x) ,

f(π)=tanπ=0[tan(π)=0]

Hence, it is proved that f(0)=f(π)

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