1. Increasing and Decreasing Functions and the First Derivative Test
3.3
Increasing and Decreasing Functions
and the First Derivative Test
A function is increasing on an interval if for any two
numbers x1 and x2 in the interval x1 < x2 implies
f(x1) < f(x2).
A function is decreasing on an interval if for any two
numbers x1 and x2 in the interval x1 < x2 implies
f(x1) > f(x2).

2. If f’(x) > 0 x in (a,b), then f is increasing on (a,b).
2. If f’(x) < x in (a,b), then f is decreasing on (a,b).
3. If f’(x) = x in (a,b), then f is constant on (a,b).
To find the open intervals on which f is increasing
or decreasing, locate the critical numbers in (a,b)
and use these numbers to determine the test intervals.
Then determine the sign of f’(x) at one value in each
of the test intervals. Use the above guidelines then to
determine where f is increasing or decreasing.

3. Ex. 1
C.N.’s 0, 1
Now, test
each interval.
inc.
dec.
inc.
1st der.
test
maximum
minimum

4. Ex. 2 2 1 2 -
dec.
inc.
dec.

5. Ex. 3
f’(-3)
< 0
f’(-1)
> 0
f’(1)
< 0
f’(3)
> 0
dec inc dec inc.
(-2, ) (0, ) (2, )
(-4)2/
1st der. test
min.
max.
min.

6. Ex. 4
C.N.’s
0, -1, 1
dec.
inc.
dec.
inc.
1st der.
test
(-1, ) (0, ) (1, ) 2 2
min
neither
min