Find the value of x that satisfies 1+3(1+3(1+3(1+x))) = 13.

The conclusion of the Mean Value Theorem says that there is a number

c

in the interval

(

1

,

3

)

such that:

f

'

(

c

)

=

f

(

3

)

−

f

(

1

)

3

−

1

To find (or try to find)

c

, set up this equation and solve for

c

.

If there's more than one

c

make sure you get the one (or more) in the interval

(

1

,

3

)

.

For

f

(

x

)

−

−

2

x

2

−

x

+

2

, we have

f

(

1

)

=

−

1

, and

f

(

3

)

=

−

18

−

3

+

2

=

−

19

Also,

f

'

(

x

)

=

−

4

x

−

1

.

So the

c

we're looking for satisfies:

f

'

(

c

)

=

−

4

c

−

1

=

f

(

3

)

−

f

(

1

)

3

−

1

=

−

19

−

−

1

3

−

1

=

−

18

2

=

−

9

So we need

−

4

c

−

1

=

−

9

. And

c

=

2

.

Note:

I hope you've been told that actually finding the value of

c

is not a part of the Mean Value Theorem.

The additional question"find the value of

c

" is intended as a review of your ability to solve equations. For most functions, you will not be able to find the

c

that the MVT guarantees us is there..

12

step-by-step explanation: