help [derivatives]?? mean value theorem?

Question

Answer 1

7)

Step 1:

The function is on the interval .

Mean Value Theorem:

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 .

.

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 .

.

Solution:

.

Answer 2

8)

Step 1:

The curve equation is .

Let the required point on the graph of is be .

Formula for the distance between two points is .

Substitute .

.

Step 2:

Write the distance formula in terms of one variable.

Consider .

Solve for .

Substitute in .

Apply derivative on each side with respect to .

.

Comments

Contd...

Step 3:

Find the extreme(minimum) value of distance function by solving .

Substitute in .

The point lies on the curve and it is closest to the fixed point is

.

Solution:

.