help-mean value theorem?

Question

Answer 1

(1)

Step 1:

The function is  , .

Rolle's theorem :

Let be a function that satisfies the following three hypotheses.

1. is continuous on .

2. is differentiable on .

3. .

Then there is a number  in  such that .

The function is continuous over and differentiable on , since it is a polynomial function.

.

Substitute in the function.

.

Substitute in the function.

Therefore  .

Hence function satisfy the Rolle's theorem.

Then there exist at least one  value in the interval ,such that .

Comments

Continued...

Step 2:

.

Differentiate on each side with respect to .

.

Equate it to zero.

Hence .

The value of c is 1.

Solution:

The value of c is 1.

Answer 2

2)

Step 1:

The function is  , . 

Rolle's theorem :

Let  be a function that satisfies the following three hypotheses.

1.  is continuous on .

2.  is differentiable on .

3. .

Then there is a number  in  such that .

The function is continuous over and differentiable on , since it is a polynomial function.

.

Substitute in the function.

.

Substitute in the function.

Therefore .

Hence the function satisfy the Rolle's theorem.

Then there exist at least one  value in the interval ,such that .

Comments

Continued...

Step 2:

.

Differentiate on each side with respect to .

.

Equate it to zero.

and .

is not in, hence it is not considered.

Hence .

The values of .

Solution:

The values of .