help? mean value theorem?

Question

Answer 1

3)
Step 1:

The function is on .

The Mean Value Thereom:

If is continuous on the closed interval   and differentiable on the open interval , then there exists a number in such that .

The function is continuous on and differentiable on .

In this case .

Step 2:

Find .

Substitute in .

f (- 3) = - 31.

Substitute in .

.

Substitute the values of and in .

.

Comments

Contd...

Step 3:

The function satisfies the mean value theorem hypotheses, then there exists a number in such that .

Apply derivative on each side with respect to .

Substitute the value of .

Solve the equation using quadratic formula.

.

Solution:

.

Answer 2

  

4)

Step 1:

The function is .

Use Intermediate Value Theorem, to show that has at least one real root.

Intermediate Value Theorem:

If is continuous on the closed interval , , and is any number between and , then there is at least one number in such that .

The polynomial function is continuous for all real numbers.

For instance the interval is [0, 1].

Substitute in .

.

Substitute in .

The function  is continuous on with and .

By Intermediate Value Theorem, there must be some in such that .

The function  has a zero in the interval .

Comments

Contd...

Step 2:

Assume the function has two real roots.

Apply derivative on each side with respect to .

By Rolle's theorem, these two roots can be obtained by solving .

Solve the equation by using quadratic formula.

.

The roots are imaginary.

This is contradictory to the statement that the function has two real roots.

Therefore, has exactly one real root.

Solution:

has exactly one real root.