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 diifferentiable on .

Step 2:

Solution:

.

4)

Step 1:

The function is .

The polynomial function is continuous over the reals.

By Intermediate Value Theorem, the function has a zero in the closed interval .

Step 2:

Assume the function has two real roots.

Apply derivative on each side with respect to .

By Rolles theorem, These two roots that there is a point where  are .

Solve the equation by using quadratic formula.

The equation has imaginary roots.

This is contraduction.

Therefore, has exactly one real root.

Solution:

has exactly one real root.