Increasing and Decreasing
Functions and the First
Derivative Test
Determine the intervals on which a function is increasing or
decreasing
 Apply the First Derivative Test to find relative extrema of a
function

Standard 4.5a
y
Constant
x
Test for Increasing and Decreasing
Functions
Let f be differentiable on the interval (a, b)
1. If f’(x) > 0 then f is increasing on (a, b)
2. If f’(x) < 0 then f is decreasing on (a, b)
3. If f’(x) = 0 then f is constant on (a, b)
Definition of Critical Number
If f is defined at c, then c is a critical number of f
if f’(c)=0 or if f’ is undefined at c.
Find the open intervals on which the given
function is increasing or decreasing.
1. Find derivative.
2. Set f’(x) = 0 and
solve to find the critical
numbers.
CRITICAL NUMBERS
3. Make table to test the sign f’(x) in each interval.
4. Use the test for increasing/decreasing to decide
whether f is increasing or decreasing on each
interval.
Interval
-∞ < x < -2
-2 < x < 2
2<x<∞
x = -3
x=0
x=3
Sign of f’(x)
f’(-3) > 0
f’(0) < 0
f’(3) > 0
Conclusion
Increasing
Decreasing
Increasing
Test value
Find the open intervals on which the given
function is increasing or decreasing.
y
Relative maximum
Relative minimum
x
Definition of Relative Extrema
Let f be a function defined at c.
1. f(c) is a relative maximum of f if there exists
an interval (a, b) containing c such that f(x) ≤
f(c) for all x in (a, b).
1. f(c) is a relative minimum of f if there exists
an interval (a, b) containing c such that f(x) ≥
f(c) for all x in (a, b).
If f(c) is a relative extremum of f, then the relative
extremum is said to occur at x = c.
1. f(c) is a relative maximum of f if there exists
an interval (a, b) containing c such that f(x) ≤
f(c) for all x in (a, b).
relative maximum
f(c)
f(x)
f(x)
f(x)
f(x)
f(x)
f(x)
f(x)
f(x)
c
2. f(c) is a relative minimum of f if there exists
an interval (a, b) containing c such that f(x) ≥
f(c) for all x in (a, b).
f(x)
f(x)
f(x)
f(x)
f(x)
f(x)
f(c)
relative minimum
Occurrence of Relative Extrema
If f has a relative minimum or a relative
maximum when x = c, then c is a critical
number of f. That is, either f’(c) = 0 or
f’(c) is undefined.
First-Derivative Test for Relative
Extrema
Let f be continuous on the interval (a, b) in which c is the
only critical number.
On the interval (a, b) if
1. f’(x) is negative to the left of x = c and positive to the
right of x = c, then f(c) is a relative minimum.
2. f’(x) is positive to the left of x = c and negative to the
right of x = c, then f(c) is a relative maximum.
3. f’(x) has the same sign to the left and right of x = c,
then f(c) is not a relative extremum.
1. f’(x) is negative to the left of x = c and positive
to the right of x = c, then f(c) is a relative
minimum.
f’(x) is
positiv
e
f’(x) is
negative
Relative minimum
c
2. f’(x) is positive to the left of x = c and
negative to the right of x = c, then f(c) is a
relative maximum.
relative maximum
f’(x) is
positiv
e
f’(x) is
negative
c
3. f’(x) has the same sign to the left and right of
x = c, then f(c) is not a relative extremum.
Not a relative
extremum
f’(x) is
positive
c
f’(x) is
positive
Find all relative extrema of the given function.
Find derivative
Set = 0 to find
critical numbers
CRITICAL NUMBERS
(-∞, -1)
(-1, 1)
(1, ∞)
x = -2
x=0
x=2
+
-
+
Increasing
Decreasing
Increasing
Relative Maximum (-1, 5)
Relative Minimum
(1, -3)
Find all relative extrema of the given function.
(-∞, -2)
(-2, 0)
(0, ∞)
x = -3
x = -1
x=1
+
-
+
Increasing
Decreasing
Increasing
Relative max: (-2, 0)
Relative min: (0, -2)
Find all relative extrema of the given function.
(0,π/4)
(π/4,3π,4)
(3π/4,5π/4)
(5π/4,7π/4)
(7π/4,2π)
x = π/6
x = π/2
x=π
x = 3π/2
x = 2π
+
-
+
-
+
Increasing
Decreasing
Increasing
Decreasing
Increasing
Relative max:
Relative min:
The graph of f is shown. Sketch a graph of the
derivative of f.
The graph of f is shown. Sketch a graph of the
derivative of f.
The graph of f is shown. Sketch a graph of the
derivative of f.