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Question

Answer 1

(1)

Step 1:

Critical number :

A critical number of a function is a number in the domain of such that either or does not exist.

The function is .

is continuous and differentiable at all values of  because it is a polynomial.

Solutions of are the critical numbers.

Differentiate on each side with respect to .

.

Step 2:

.

Equate to zero.

.  

Critical number is  .

Solution:

Critical number is  .

Answer 2

(2)

Step 1:

Critical number :

A critical number of a function is a number in the domain of such that either or does not exist.

The function is .

The domain of a function is all values of , those makes the function mathematically correct.

There should not be any negative number in the square root.

.

The domain of the function is .

Step 2:

Differentiate on each side with respect to .

Apply product rule in derivatives: .

.

Comments

Contd...

Step 3:

.

Equate to zero.

.

is not defined at .

is in the domain of .

The critical points are and .

Solution:

The critical points are and .

Answer 3

(3)

Step 1:

The function is , on the interval .

Evaluate the critical points.

The function is .

Differentiate on each side with respect to .

.

Find the critical points, by equate to zero.

The critical point is .

Step 2:

Absolute extrema of a function exist either at the end points or at the critical points.

Substitute the critical point in the function.

Substitute in .

.

Comments

Contd...

Step 3:

Evaluate function at the end points.

The function is on the interval .

Substitute in .

.

Substitute  in .

.

The maximum value of the function is at .

The absolute maximum is .

The minimum value of the function is at .

The absolute minimum is .

Solution:

The absolute maximum is .

The absolute minimum is .

Answer 4

(4)

Step 1:

The function is , on the interval .

Evaluate the critical points.

The function is .

Differentiate on each side with respect to .

.

Find the critical points, by equate to zero.

Solution of the equation in the interval is .

The critical point is .

Step 2:

Absolute extrema of a function exist either at the end points or at the critical points.

Substitute the critical point in the function.

Substitute in .

.

Comments

Contd...

Step 3:

Evaluate function at the end points.

The function is on the interval .

Substitute in .

.

Substitute  in .

.

The maximum value of the function is at .

The absolute maximum is .

The minimum value of the function is at  .

The absolute minimum is .

Solution:

The absolute maximum is .

The absolute minimum is .

Answer 5

(5)

Step 1:

The function is  and the point is .

Apply derivative on each side with respect to .

.

Find the slope of a tangent at the point .

Substitute in .

Slope of a tangent line is .

Step 2:

Find the tangent line equation.

Point - slope form of line equation is .

Substitute the values and in point slope form.

.

The tangent line equation is .

Solution:

The tangent line equation is .