Chapter 1.

1.1 Problem Statement

{2,4,6,8}
{1,3,5}
$p + $p1
-1*$p
-1*$q
$p + $q
$p*$q
rand(2,9)
$pmax - 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).

1.2 Step 1

Question Sequence

Question 1.1

When f'(x) > 0 for all x on a defined open interval, then f is zfi1I3d4PvPpiEdCcEYd3mwzJ2Jn7ZmnBM/VedVTNWNmHo9/C0wmIKERkIE= on that interval.

Correct.
Incorrect.

Question 1.2

When f'(x) < 0 for all x on a defined open interval, then f is lb1yDssyhRoDeV34+lBpA1Q20o6SPKA4XGI914rrm6KYpxdQB7vRY8wbTYQ= on that interval.

Correct.
Incorrect.

Question 1.3

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) = 1Wh3cvJ2xF4= or where f'(c)risoekzpYAAgRXECVAcsXWXnXFT9zJ4gDoyL7g==.

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.4

Find f'(x).

where b = nc1ItEz0kR4= and c = iSba6t70dtA=.

Correct.
Incorrect.

Question 1.5

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

The critical points are

c =VuqerCroNnN8kEVC, JRTGa0xCnPWOxQpY.

(Enter the smaller value first.)

Correct.
Incorrect.

1.4 Step 3

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

(-∞,$pmax),($pmax,$pmin),($pmin,∞)

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 1.6

Let's pick x1 = $x1 in the first interval, x2 = $pmax 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)
(-∞,$pmin) $x1 brSjF5lOKMQ=
($pmin,$pmax) $pmax 097XeLvBC5c=
($pmax,∞) 0 brSjF5lOKMQ=
Correct.
Incorrect.

Question 1.7

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

H9aHpkMf02hQ4OjDixAlXGOJkScAubK24CWmXA== (-∞,$pmin) ($pmax,∞)

p2CFj3VtDXt2cxcALfSQpMJFT8s4YoKX6KzyhQ== ($pmax,$pmin)

Correct.
Incorrect.

1.5 Step 4

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

Question Sequence

Question 1.8

If f'(x) changes sign from + to − at x = c, then f(c) is a local MLrjsAZLmRw2xfA/t8djnqWlqT8=.

If f'(x) changes sign from − to + at x = c, then f(c) is a local cCmYpIf3+uu6kk3FVOz/JXh4+jw=.

Correct.
Incorrect.

Question 1.9

Thus, f has a local minimum at c = VuqerCroNnN8kEVC and a local minimum at c = JRTGa0xCnPWOxQpY.

Correct.
Incorrect.