# To find the interval at which the function f is increasing.

### 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 5.4, Problem 21E
To determine

## To find the interval at which the function f is increasing.

Expert Solution

The function f is increasing at interval (1,1) .

### Explanation of Solution

Given information:

The function is f(x)=0x(1t2)et2dt

Concept used:

The fundamental theorem of calculus, part 1 is defined by,

If f is continuous on [a,b] , then the function g defined by,

g(x)=axf(t)dt

axb is an antiderivative of f , that is g(x)=f(x) for a<x<b .

Let h(t)=(1t2)et2 ,

f(x)=0xh(t)dt

Using Part 1 of the Fundamental Theorem of Calculus,

f(x)=h(x) [since, h(t) is continuous and 0<x<b ]

Therefore,

The critical point of the function f is obtained as:

f(x)=h(x)=(1x2)ex2

Put, f(x)=h(x)=0

(1x2)ex2=0

Either,

1x2=0x2=1x=±1

Now, the function is said to be increasing if f(x)>0 ,

Since, (1x2) will remain positive at interval 1<x<1 and ex2 is positive for all values of x .

Therefore,

The function f is increasing at interval (1,1) .

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