Section 3.3 – Increasing and
Decreasing Functions and the
First Derivative Test
How Derivatives Affect the Shape of the
Graph
Many of the applications of calculus depend
on our ability to deduce facts about a
function f from in information concerning
its derivatives. Since the derivative of f
represents the slope of tangents lines, it
tells us the direction in which the curve
proceeds at each point. Thus, it should
seem reasonable that the derivative of a
function can reveal characteristics of the
graph of the function.
Increasing and Decreasing Functions
The function f is strictly increasing on an
interval I if f (x1) < f (x2) whenever x1 < x2.
The function f is strictly Decreasing on an
interval I if f (x1) > f (x2) whenever x1 < x2.
D
f(x)
Decreasing
Increasing
B
A
C
x
How the Derivative is connected to
Increasing/Decreasing Functions
When the function is increasing, what is the
sign (+ or –) of the slopes of the tangent
lines? POSITIVE Slope
When the function is decreasing, what is the
sign (+ or –) of the slopes of the tangent
lines? NEGATIVE Slope
D
f(x)
B
A
C
x
Test for Increasing and Decreasing
Functions
Let f be differentiable on the open interval (a,b)
If f '(x) > 0 on (a,b), then f is strictly increasing
on (a,b).
If f '(x) < 0 on (a,b), then f is strictly decreasing
on (a,b).
If f '(x) = 0 on (a,b), then f is constant on (a,b).
Procedure for Finding Intervals on which
a Function is increasing or Decreasing
If f is a continuous function on an open interval (a,b). To
find the open intervals on which f is increasing or
decreasing:
1. Find the critical numbers of f in (a,b) AND all values
(a,b) of x in that make the derivative undefined.
2. Make a sign chart: The critical numbers and x-values
that make the derivative undefined divide the x-axis
into intervals. Test the sign (+ or –) of the derivative
inside each of these intervals.
3. If f '(x) > 0 in an interval, then f is increasing in that
same interval. If f '(x) < 0 in an interval, then f is
decreasing in that same interval.
4. State your conclusion(s) with a “because” statement
using the sign chart.
A sign chart does NOT stand on its own.
Example 1
Use the graph of f '(x) below to determine when f is
increasing and decreasing.
f is
increasing
when the
derivative is
positive.
f ' (x)
Increasing: (-∞,-1) U (3,∞)
f is
increasing
when the
x
derivative is
f is positive.
decreasing
when the
derivative is
negative.
Decreasing: (-1,3)
White Board Challenge
The graph of f is shown below. Sketch a graph
of the derivative of f.
White Board Challenge
Find the critical numbers of:
f  x   2 x  9 x  12 x
3
2
x  2 or
x 1
Example 2
4
3
2
Find where the function f  x   3x  4 x  12 x  5 is
Domain of f:
increasing and where it is decreasing.
All Reals
Find the critical numbers/where the derivative is undefined
Find the derivative.
f ' x 
d
dx
4
3
2
3
x

4
x

12
x
 5

Find where the derivative is 0 or undefined
0  12 x3  12 x 2  24 x
0  12 x  x 2  x  2 
f '  x   12 x3 12x2  24x
0  12 x  x  2 x  1
Find the sign of the derivative on each
x  0, 2,  1

x  2

-1
interval.
x  0.5
0


2
f ' x
x 1
x3
Answer the question
f ' 1  24
f '  2  96
f '  0.5  7.5
f '  3  144
Example 2: Answer
The function is increasing on
(-1,0)U(2,∞) because the first
derivative is positive on this
interval.
The function is decreasing on
(-∞,-1)U(0,2) because the first
derivative is negative on this
interval.
Example 3
Use the graph of f (x) below to determine when f is
increasing and decreasing.
f is decreasing
Notice how the
function changes
from increasing to
f is increasing
decreasing
at x=-3.
But when
since the
-3 is not
function’s
outputs
in the domain
of
are
larger
the getting
function,
it is
the inputpoint.
notas
a critical
increases.
Thus,
critical points
are not the only
points to include in
sign charts.
Increasing: (-∞,-3)
when the
function’s outputs
are getting
smaller as the
input increases.
f (x)
x
Decreasing:(-3,∞)
Example 4
Find where the function f  x   x11 is increasing and where
it is decreasing.
Domain of f : All Reals except -1
Find the critical numbers/where the derivative is undefined
Find the derivative.
f ' x 
 x 1 dxd 11 dxd  x 1
 x 12
f ' x  
1
 x 12
Find where the derivative is 0 or undefined
The function does not have any critical points:
the derivative is never equal to 0 and the
derivative is only undefined at a point not in the
functions domain (x=-1).
Even though -1 is not a critical point, it can still be a point where a function
changes from increasing to decreasing. ALWAYS include every x value that
makes the derivative undefined on a sign chart (even if its not a critical point).
Find the sign of the derivative on each interval.

x  2
f '  2   1

-1
f '  0   1
f ' x
Answer the question
Example 4: Answer
The function is decreasing on
(-∞,-1)U(-1,∞) because the first
derivative is negative on this
interval.
Make sure not to include -1 in the interval because it is not
in the domain of the function.
White Board Challenge
Find the maximum and minimum values
attained by the given function on the
indicated closed interval:
f  x  x  ;
4
x
1, 4
max : 5 @ x  1, 4
min : 4 @ x  2
Critical Values and Relative Extrema
Remember that if a function has a relative
minimum or a maximum at c, then c must be
a critical number of the function.
Unfortunately not every critical number
results in a relative extrema.
A new calculus method is needed to determine
whether relative extrema exist at a critical
point and if it is a maximum or minimum.
How the Derivative is connected to
Relative Minimum and Maximum
When a critical point is a relative maximum, what are the
characteristics of the function?
The function changes from increasing to decreasing
When a critical point is a relative minimum, what are the
characteristics of the function?
The function changes from decreasing to increasing
D
f(x)
B
A
C
x
The First Derivative Test
Suppose that c is a critical number of a continuous function
f(x).
(a) If f '(x) changes from positive to negative at c, then f(x)
has a relative maximum at c.
f(x)
Relative Maximum
f '(x) > 0
f '(x) < 0
c
x
The First Derivative Test
Suppose that c is a critical number of a continuous function
f(x).
(b) If f '(x) changes from negative to positive at c, then f(x)
has a relative minimum at c.
f(x)
f '(x) < 0
f '(x) > 0
Relative Minimum
c
x
The First Derivative Test
Suppose that c is a critical number of a continuous function
f(x).
(c) If f '(x) does not change sign at c (that is f '(x) is positive
on both sides of c or negative on both sides), then f(x)
has no relative maximum or minimum at c.
f(x)
f(x)
f '(x) > 0
f '(x) > 0
c
x
No
Relative
Maximum
or
Minimum
f '(x) < 0
f '(x) < 0
c
x
Example 1
Use the graph of f '(x) below to determine where f has a
relative minimum or maximum.
f ' (x)
Find the Critical Numbers
Make a sign chart and Find the
sign of the derivative on each
interval.
x
Apply the First Derivative Test.


x  2
Relative Maximum: @ -1
-1
x0

3
f ' x
x4
Relative Minimum: @ 3
Example 2
Find where the function f  x   x  2sin x on 0≤x≤2π has
Domain of f:
relative extrema.
0≤x≤2π
Find the critical numbers
Find the derivative.
f '  x   dxd  x  2sin x 
Find where the derivative is 0 or undefined
f '  x   1  2cos x
Find the sign of the derivative on each
interval.


0
x 1
2π/3
x3

4π/3
f ' x
2π
x5
f '  5  1.57
f ' 1  2.08
f '  3  0.98
0  1  2cos x
1  2cos x
 12  cos x
x  23 , 43
Find the value of the function:
f  23   3  23
f  43   43  3
Answer the question
Example 2: Answer
The function has a relative maximum
of 3.826 at x = 2π/3 because the first
derivative changes from positive to
negative values at this point.
The function has a relative minimum
2.457 at x = 4π/3 because the first
derivative changes from negative to
positive values at this point.
Example 3
13
Find the relative extrema values of f  x   x  x  3 .Domain of f:
All Reals
Find the critical numbers
23
Find the derivative.
0  3x 
 x  3 
23
1 3
1 2 3
2 13
 x  3
f '  x   3 x  x  3  3 x  x  3
2x


3x
3 x  3
 x  3
2x
f '  x   3 x  3 x 3
 
3  x  3  6 x
f ' x 
d
dx
x
Find where the derivative is 0 or undefined
 x  32 3
23
13
23
23
23
23
13
23
13
Find the sign of the derivative
on each interval.


x  4
-3
x  2

-1
0
2 x1 3
13
3 x  3
13
13
The
derivative
is
undefined
at x=-3,0
3x  9  6x
9  9x
x  1

f ' x
NOTE: 0 is not a relative
x 1
x  0.5
extrema since the derivative
f '  0.5  0.58
f '  4  1.19
does not change sign.
f '  2  0.63
f ' 1  1.26 Answer the question
Example 3: Answer
First find the value of the function:
f  3  0
23
f  1  2
Now answer the question:
The function has a relative maximum of 0 at
x = -3 because the first derivative changes
from positive to negative values at this point.
The function has a relative minimum of
-1.587 at x = -1 because the first derivative
changes from negative to positive values at
this point.
White Board Challenge
Find the intervals on which the function below
is increasing or decreasing.
f  x   1  x  x
3
BONUS: How
many critical
numbers are
there?
increasing : x 
decreasing : x 
1
4
1
4