# Whether the statement, if the function f is increasing and f ( x ) &gt; 0 on I then g ( x ) = 1 f ( x ) is decreasing on I 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 15RQ
To determine

## Whether the statement, if the function f is increasing and f(x)>0 on I then g(x)=1f(x) is decreasing on I is true or false.

Expert Solution

The given statement is true.

### Explanation of Solution

If a function f is increasing then f(x)>0 for all values of x

Also it is given that f(x)>0 .

Show that (g)<0 for all values of x.

It is given that g(x)=1f(x) .

The derivative of g(x) is,

g(x)=ddx(1f(x))=f(x)f2(x)

As f2(x)>0 and f(x)>0 deduce that, g(x)<0 .

Since, g(x)<0 the function g(x)=1f(x) is an decreasing function.

Therefore, the given statement is true.

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