# Whether the statement, there exist a function f such that f ( x ) &gt; 0 , f ′ ( x ) &lt; 0 and f ′ ′ ( x ) &gt; 0 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 9RQ
To determine

## Whether the statement, there exist a function f such that f(x)>0,f′(x)<0 and f′′(x)>0 for all x  is true or false.

Expert Solution

The given statement is true.

### Explanation of Solution

Consider a function in such a way that f(x)>0 . So, consider f(x)=1x .

Differentiate f with respect to x.

f(x)=1x2

Thus, f(x)<0 for all x.

Find the second derivative of f(x) .

f(x)=2xx4=2x3

The second derivative function, f(x)>0 for all x.

Therefore, the given statement is true.

### Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!