Chapter 1.

1.1 Problem Statement

{2,4,6,8}
{3,5,7}
$a/2
$a*$b
3*$a
6*$a
$a/4
$a2 + 1

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

f(x) = (x - $a)($b - x3)

1.2 Step 1

Question Sequence

Question 1.1

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 mJmoTZdFvETWOfkSuPQvC7zt+Jq9To11LFxThg== in that interval.

If f"(x) < 0, then f(x) is WZxdTiKTzgf4WeM8lZHGyjbyPec+0DBl4QnenQ== in that interval.

Correct.
Incorrect.

Question 1.2

Find f'(x):

f(x) = (x - $a)($b - x3)

f'(x) = sWNiRN4ZHUU=·x3 + j8gjfTtT5bc=·x2 + iSba6t70dtA=

Correct.
Incorrect.

Question 1.3

Find f''(x):

f''(x) = DDH6Tw1RFEk=·x (uEwXrt+3HqA= - x)

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.4

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 Kgz5Jqb4BpmCwJeIJdDFsJ+ZtGGPC5dPgQEb0nMZsBcZ7j+cv298R0cHuQOv0HlRlemNqw== so that we can then determine any change in concavity at x = c.

Correct.
Incorrect.

Question 1.5

Find any potential points of inflection at x = c.

f''(x) = 12·x ($a2 - x)

c = 1Wh3cvJ2xF4= (smaller value)

c = uEwXrt+3HqA= (larger value)

Correct.
Incorrect.

1.4 Step 3

Use c = 0 and c = $a2 to divide the real line into three intervals where we will determine the sign of f"(x).

(-∞,0), (0,$a2), ($a2,∞)

Question Sequence

Question 1.6

Let's pick x1 = −1 in the first interval, x2 = $a4 in the second interval, and x3 = $a2plus1 in the third interval. Fill in the table below with the appropriate sign.

Interval x-value Sign of f''(x)
(-∞,0) -1 097XeLvBC5c=
(0,$a2) $a4 brSjF5lOKMQ=
(0,∞) $a2plus1 097XeLvBC5c=
Correct.
Incorrect.

Question 1.7

The points of inflection are the following points.

x = 1Wh3cvJ2xF4= (smaller value)

x = uEwXrt+3HqA= (larger value)

Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.8

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

f(x) is mJmoTZdFvETWOfkSuPQvC7zt+Jq9To11LFxThg== on the interval (0,$a2).

f(x) is +fjpoHM3bomWG6vz3DKIar00+kxXtgRrRBwWtg== on the intervals (-∞,0) ($a2,∞).

Correct.
Incorrect.