   Chapter 4.2, Problem 24E ### 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

# (a) Suppose that f is differentiable on ℝ and has two roots. Show that f′ has at least one root.(b) Suppose f is twice differentiable on ℝ and has three roots. Show that f″ has at least one real root.(c) Can you generalize parts (a) and (b)?

(a)

To determine

To show: The function f has at least one root if f is differentiable on and has two roots.

Explanation

Theorem used: Rolle’s Theorem

“If 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

(b)

To determine

To show: The function f has at least one root if f is twice differentiable on and has three roots.

(c)

To determine

To compare: part (a) and part (b) and generalize the conclusion.

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