   Chapter 4.2, Problem 27E ### 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

# Does there exist a function f such that f(0) = −1, f(2) = 4, and f′(x) ≤ 2 for all x?

To determine

To check: Whether the function exists or not for the given condition that is f(0)=1 , f(2)=4 and f(x)2 for all x.

Explanation

Theorem used: Mean value Theorem

“If 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) ”.

Reason:

Suppose that such function f exists, then by the Mean value Theorem mentioned above, there exist a number c in the interval (0,2) such that, f(c)=f(2)f(0)20

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