5.3 - Determining Intervals on Which a Function Is Increasing or Decreasing

Critical Points

A critical point happens when either:

The derivative equals zero: $f'(x) = 0$

The derivative doesn’t exist

5.3 - Determining Intervals on Which a Function Is Increasing or Decreasing

Between critical points, a function is either entirely increasing or entirely decreasing. It can’t switch behavior without passing through a critical point, or through a break in the graph like a vertical asymptote. So any

x

-values where

f

is undefined go in your sign chart too (see 5.4, Example 2).

Sign Analysis Method

Find the derivative $f'(x)$

Find the critical points by solving $f'(x) = 0$ and checking where $f'(x)$ doesn’t exist

Create intervals using the critical points as boundaries

Test the sign of $f'(x)$ in each interval

Write your conclusions with proper justification

Example 1

Given

f(x) = -x^2 - 4x + 7

Step 1: Find the derivative.

f'(x) = -2x - 4

Step 2: Find critical points.

-2x - 4 = 0 \quad \Rightarrow \quad x = -2

Step 3: Build a sign chart.

Interval	$(-\infty, -2)$	$x = -2$	$(-2, \infty)$
Sign of $f'(x)$	$+$	$0$	$-$

Step 4: Test the signs.

For $x < -2$ , test $x = -3$ : $f'(-3) = -2(-3) - 4 = 2 > 0$

For $x > -2$ , test $x = 0$ : $f'(0) = -4 < 0$

Step 5: Conclusions.

f

is increasing on

(-\infty, -2)

because

f'(x) > 0

f

is decreasing on

(-2, \infty)

because

f'(x) < 0

Example 2

Given

h(x) = x^3 - 3x^2 - 45x + 10

Find the derivative:

h'(x) = 3x^2 - 6x - 45

Factor:

h'(x) = 3(x^2 - 2x - 15) = 3(x - 5)(x + 3)

Critical points:

x = -3

and

x = 5

Sign chart:

Interval	$(-\infty, -3)$	$x = -3$	$(-3, 5)$	$x = 5$	$(5, \infty)$
Sign of $h'(x)$	$+$	$0$	$-$	$0$	$+$

The factored form makes sign testing easy:

$x = -4$ : $(-4 - 5)(-4 + 3) = (-)(-) = (+)$

$x = 0$ : $(0 - 5)(0 + 3) = (-)(+) = (-)$

$x = 6$ : $(6 - 5)(6 + 3) = (+)(+) = (+)$

Conclusions:

h

is increasing on

(-\infty, -3) \cup (5, \infty)

because

h'(x) > 0

h

is decreasing on

(-3, 5)

because

h'(x) < 0

Reading from the Graph of $f'(x)$

When you’re given the graph of the derivative, you can figure out where the original function is increasing or decreasing.

Where

f'(x) > 0

(above the x-axis),

f

is increasing: on

(-\infty, -3)

and

(1, 3)

Where

f'(x) < 0

(below the x-axis),

f

is decreasing: on

(-3, 1)

and

(3, \infty)

Justification:

f

increases where

f'(x) > 0

and decreases where

f'(x) < 0

Applications: Rates of Change

The sign of a rate of change tells you whether the dependent variable is going up or down.

\text{Rate of change} = \frac{\Delta y}{\Delta x}

If the rate is positive, the quantity is increasing

If the rate is negative, the quantity is decreasing

Interpretations

Rate	Sign	What it means
students/year	$(+)$	The number of students is increasing
miles/hour	$(-)$	The number of miles is decreasing
cases/week	$(+)$	The number of cases is increasing
bacteria/(day²)	$(-)$	The rate of bacteria per day is decreasing

Note: units with squared time (like “per day squared”) represent the rate of change of a rate, similar to a second derivative.

Application Problem

Problem: The rate of change of bacteria in a petri dish at time

t

hours is modeled by

B(t) = 15e^{0.3t}\cos(t) \text{ bacteria per hour}

Show that the number of bacteria is decreasing at

t = 4

hours.

Solution: The problem hands you

B(t)

, which is already the rate of change. So to show the bacteria count is decreasing, you just need

B(4) < 0

B(4) = 15e^{1.2}\cos(4) \approx 15(3.3201)(-0.6536) \approx -32.55

The number of bacteria is decreasing at

t = 4

because

B(4) \approx -32.55 < 0

Common error: Do NOT find

B'(4)

. That gives you the rate of change of the rate, which isn’t what the question asked.

Key Question Pattern

When a problem says “the rate of ____ is modeled by

R(t)

”:

To show the quantity is increasing or decreasing, check the sign of $R(t)$

To show the rate is increasing or decreasing, check the sign of $R'(t)$

Read carefully and figure out what the given function actually represents.

Key Takeaways

Critical points happen where $f'(x) = 0$ or $f'(x)$ doesn’t exist

Between critical points (and breaks in the graph), a function is entirely increasing or entirely decreasing

$f$ is increasing where $f'(x) > 0$

$f$ is decreasing where $f'(x) < 0$

Use the factored form of the derivative to make sign testing easier

Given the graph of $f'(x)$ , just look at where it’s above or below the x-axis

Always justify: “because $f'(x) > 0$ ” or “because $f'(x) < 0$ ”

For rate problems, figure out whether the given function is the quantity or its rate

Back Next

5.3 - Determining Intervals on Which a Function Is Increasing or Decreasing

Critical Points

Sign Analysis Method

Example 1

Example 2

Reading from the Graph of f′(x)f'(x)f′(x)

Applications: Rates of Change

Interpretations

Application Problem

Key Question Pattern

Key Takeaways

Reading from the Graph of $f'(x)$