# Whether the statement “There exists a function f such that f ( 1 ) = − 2 , f ( 3 ) = 0 and f ′ ( x ) &gt; 1 for all x ” is true or false.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4, Problem 8RQ
To determine

Expert Solution

## Answer to Problem 8RQ

The given statement is false.

### Explanation of Solution

Result used:

Mean value theorem:

If f is a continuous function on [a,b] such that f is differentiable on (a,b), then there exits a point c in (a,b) such that f(c)=f(b)f(a)ba

Calculation:

Given statement is false since the following example disproves it.

Consider the two points (1,f(1)) and (3,f(3)).

The slope of the line joining these points is given by

f(3)f(1)31=0(2)2=1

Note that the slope represents the first derivative of the function.

Thus, there exists some point c in (1,3) for which the value of first derivative is 1.

Therefore f(x)>1 for all x is not possible.

Therefore, the given statement is false.

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