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Increasing and Decreasing
Functions
AP Calculus – Section 3.3
Increasing and Decreasing Functions
On an interval in which a function f is
continuous and differentiable, a function
is…
 increasing if f ‘(x) is positive on that
interval,
 decreasing if f ‘(x) is negative on that
interval, and
 constant if f ‘(x) = 0 on that interval.
Visual Example
f ‘(x) < 0 on (-5,-2)
f(x) is decreasing on (-5,-2)
f ‘(x) = 0 on (-2,1)
f(x) is constant on (-2,1)
f ‘(x) > 0 on (1,3)
f(x) is increasing on (1,3)
Finding Increasing/Decreasing
Intervals for a Function
To find the intervals on which a function is
increasing/decreasing:
1. Find critical numbers.
2. Pick an x-value in each closed interval
between critical numbers; find derivative
value at each.
3. Test derivative value tells you whether
the function is increasing/decreasing on
the interval.
Example
Find the intervals on which the function
3
f ( x)  x  x is increasing and decreasing.
2
3
2
Critical numbers:
f ' ( x)  3 x 2  3 x
3x 2  3x  0
3 x( x  1)  0
x  {0,1}
Example
Test an x-value in each interval.
Interval
Test Value
f ‘(x)
(,0)
(0,1)
(1,  )
1
1
2
2
f ' ( 1)  6
3
1
f '   
4
2
f(x) is increasing on (,0and
)
f(x) is decreasing on (0.,1)
f ' ( 2)  6
(1,. )
Assignment
p.181: 1-5, 7, 9
The First Derivative Test
AP Calculus – Section 3.3
The First Derivative Test
If c is a critical number of a function f, then:
 If f ‘(c) changes from negative to positive
at c, then f(c) is a relative minimum.
 If f ‘(c) changes from positive to negative
at c, then f(c) is a relative maximum.
 If f ‘(c) does not change sign at c, then f(c)
is neither a relative minimum or
maximum.
 GREAT picture on page 176!
Find all intervals of increase/decrease and
all relative extrema.
f ( x)  x 2  8x  10
f ' ( x)  2 x  8
2x  8  0
x  4
f is decreasing
before -4 and
increasing after -4;
so f(-4) is a
MINIMUM.
Test:
(,4)
f ' (5)  2(5)  8  2
f is decreasing
Test:
(4, )
f ' ( 0)  8
f is increasing
Assignment
p.181: 11-21 odd