Problem Statement

{2,4,6,8}
{3,5,7}
4/2
4*5
3*4
6*4
4/4
2 + 1

Determine the intervals on which the function is concave up or concave down and find the points of inflection.

f(x) = (x - 4)(5 - x3)

 
Step 1

Question Sequence

Question 1

The second derivative Test for Concavity states that when f''(x) exists for all x in an open interval:

If f"(x) > 0, then f(x) is in that interval.

If f"(x) < 0, then f(x) is in that interval.

Correct.
Incorrect.

Question 2

Find f'(x):

f(x) = (x - 4)(5 - x3)

f'(x) = ·x3 + ·x2 +

Correct.
Incorrect.

Question 3

Find f''(x):

f''(x) = ·x ( - x)

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 4

A point P = (c, f(c)) is a point of inflection of f(x) if the concavity changes from up to down or from down to up at x = c. Thus, we need to find where f''(c) is so that we can then determine any change in concavity at x = c.

Correct.
Incorrect.

Question 5

Find any potential points of inflection at x = c.

f''(x) = 12·x (2 - x)

c = (smaller value)

c = (larger value)

Correct.
Incorrect.

 
Step 3

Use c = 0 and c = 2 to divide the real line into three intervals where we will determine the sign of f"(x).

(-∞,0), (0,2), (2,∞)

Question Sequence

Question 6

Let's pick x1 = −1 in the first interval, x2 = 1 in the second interval, and x3 = 3 in the third interval. Fill in the table below with the appropriate sign.

Interval x-value Sign of f''(x)
(-∞,0) -1
(0,2) 1
(0,∞) 3
Correct.
Incorrect.

Question 7

The points of inflection are the following points.

x = (smaller value)

x = (larger value)

Correct.
Incorrect.

 
Step 4

Question Sequence

Question 8

Based on the sign of f''(x) in the table:

f(x) is on the interval (0,2).

f(x) is on the intervals (-∞,0) (2,∞).

Correct.
Incorrect.