Problem Statement

{2,4,6,8}
{1,3,5}
2 + 3
-1*2
-1*5
2 + 5
2*5
rand(2,9)
-5 - 1

Find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

 
Step 1

Question Sequence

Question 1

When f'(x) > 0 for all x on a defined open interval, then f is on that interval.

Correct.
Incorrect.

Question 2

When f'(x) < 0 for all x on a defined open interval, then f is on that interval.

Correct.
Incorrect.

Question 3

In order to determine intervals of increase and decrease, we need the critical points of f(x).

A number c in the domain of f is called a critical point if either f'(c) = or where f'(c).

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 4

Find f'(x).

where b = and c = .

Correct.
Incorrect.

Question 5

Solve f'(x) = 0 to find any critical points, c.

The critical points are

c =, .

(Enter the smaller value first.)

Correct.
Incorrect.

 
Step 3

Use the two critical points to divide the real line into three intervals.

(-∞,-5),(-5,-2),(-2,∞)

To determine where f(x) is increasing or decreasing, pick an x-value in each interval, and identify the sign of f'(x).

Question Sequence

Question 6

Let's pick x1 = -6 in the first interval, x2 = -5 in the second interval, and x3 = 0 in the third interval. Fill in the table below with the appropriate sign.

Interval x-value Sign of f'(x)
(-∞,-2) -6
(-2,-5) -5
(-5,∞) 0
Correct.
Incorrect.

Question 7

Thus, f is increasing and decreasing on the following interval(s).

(-∞,-2) (-5,∞)

(-5,-2)

Correct.
Incorrect.

 
Step 4

The First Derivative Test for critical points states that for any critical point x = c:

Question Sequence

Question 8

If f'(x) changes sign from + to − at x = c, then f(c) is a local .

If f'(x) changes sign from − to + at x = c, then f(c) is a local .

Correct.
Incorrect.

Question 9

Thus, f has a local minimum at c = and a local minimum at c = .

Correct.
Incorrect.