Step 1:

The function is , .

Mean value theorem :

Let f be a function that satisfies the following three hypotheses :

1. f is continuous on 

2. f is differentiable on 

Then there is a number c in  such that  .

Step 2:

The function is .

The function is continuous on the interval .

Differentiate  with respect to .

\ \

The function is differentiable on the interval .

Then .

Step 3:

From the mean value theorem :

.

Substitute in .

General solution of is .

If then .

If then .

are not in the interval , hence they are not considered. \ \

Solution:

.

Step 1:

(b)

The function is , .

Mean value theorem :

Let f be a function that satisfies the following three hypotheses :

1. f is continuous on 

2. f is differentiable on 

Then there is a number c in  such that  .

Step 2:

The function is .

The function is continuous on the interval .

Differentiate  with respect to .

\ \

The function is differentiable on the interval .

Then .

Step 3:

From the mean value theorem :

.

Substitute in .

are not in the interval , hence it is not considered. \ \

Solution:

.