# Whether the statement, there exist a function f such that f ( x ) &lt; 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 10RQ
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 false.

### Explanation of Solution

Results used:

Concavity test:

(a)If a function is such that f(x)>0 then the graph of f is concave upward.

(b) If a function is such that f(x)<0 then the graph of f is concave upward.

It is given that f(x)<0 thus graph of f is always below the x-axis.

Also, as f(x)<0 the function is a decreasing function.

From concavity test, as f(x)>0 the graph of the function is concave upward.

If the graph is concave upward then the graph must cross the x-axis at some point as x approaches infinity.

This implies that the function is positive. So, this is not possible.

Hence, the given statement is false.

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