# The graph of afunctionwherevalues of function are given at some points. ### 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 2.6, Problem 21E
To determine

## To sketch:The graph of afunctionwherevalues of function are given at some points.

Expert Solution

f(x)=x332x2+3x .

### Explanation of Solution

Given: y=f(x) be any function.

f(0)=0,f'(0)=3,f'(1)=0, and f'(2)=1

Let f'(x)=ax2+bx+c be any function.

It has to sketch the graph where some values are given.

Giventhatf'(0)=3

Therefore

3=a.02+b.0+c3=c(1)

Giventhatf'(1)=0

Therefore

0=a.12+b.1+c0=a+b+c

Using equation (1) it can substitute c=3

0=a+b+3

a+b=3(2)

Giventhatf'(2)=1

Therefore

1=a.22+b.2+c-1=4a+2b+c

Using equation (1) it can substitute c=3

-1=4a+2b+3

4a+2b=4

Dividing both side by 2

2a+b=2(3)

Adding equation (1) and equation (2) , to get

a+b2=3+2a+b

Subtract a+b from both side

2=3+a

1=a

Substitute a=1 in equation (2)

1+b=3b=4

Substitute the value of to get a,b  and c

f'(x)=x24x+3

Integrate to get

f(x)=x32x2+3x+c

Given that f(0)=0

Therefore,

f(0)=0332.02+3.0+cc=0

f(x)=x332x2+3x

Now graph of this function is given below Hence, the is functionis f(x)=x332x2+3x .

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