Objectives:
1. Be able to determine where a function is increasing,
decreasing or constant with the use of calculus.
Critical Vocabulary:
Increasing, Decreasing, Constant
Warm Up: Use the graphs to determine where the graph is
increasing, decreasing, or constant.
[answers should be in interval notation]
Time to look at this from a Calculus perspective.
What kind of relationship exists between the intervals
that are increasing and deceasing and the
corresponding slopes in these regions?
A function f is increasing on an interval for any 2 numbers
x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2).
Since x1 < x2 (1 < 2) that implies f(x1) < f(x2) (0 < 3)
A function f is increasing on an interval for any 2 numbers
x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2).
A function f is decreasing on an interval for any 2 numbers
x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2).
Since x1 < x2 (-6 < -4) that implies f(x1) > f(x2) (3 > -3)
How to test for Increasing and Decreasing using CALCULUS
1. Locate the critical numbers to determine our
boundaries for our test intervals.
2. Determine the sign of f’(x) at 1 test value in each
interval.
3. Use the following theorem to determine if that
interval is increasing or decreasing.
If f is continuous on the closed interval [a, b] and
differentiable on the open interval (a, b):
1. If f’(x) > 0 for all x in (a, b), then f is increasing on [a,b]
2. If f’(x) < 0 for all x in (a, b), then f is decreasing on [a,b]
3. If f’(x) = 0 for all x in (a, b), then f is constant on [a,b]
Example 1: Find the open intervals on which the
following function is increasing or
decreasing.
3
f ( x)  x 3  x 2
2
1st: Find the critical numbers
Increasing: (-∞, 0)(1, ∞)
f’(x) = 3x2 - 3x
Decreasing: (0, 1)
0 = 3x2 - 3x
0 = 3x(x - 1)
Constant: Never
x = 0 and x = 1
2nd: Test Values
Interval
Test Value
(-∞, 0)
x = -5
(0, 1)
x=½
(1, ∞)
x = 25
Sign of f’(x) f’(x) = 90 (+) f’(x) = -3/4 (-) f’(x) = 1800 (+)
Conclusion
Increasing
Decreasing
Increasing
Example 1: Find the open intervals on which the
following function is increasing or
decreasing.
3
f ( x)  x 3  x 2
2
Increasing: (-∞, 0)(1, ∞)
Decreasing: (0, 1)
Constant: Never
Example 2: Find the open intervals on which the
following function is increasing or
decreasing.
x2
f ( x) 
x 1
1st: Find the critical numbers
Increasing: (-∞, -2)(0, ∞)
x2  2x
f ' ( x) 
( x  1) 2
Decreasing: (-2, -1) (-1, 0)
0 = x2 + 2x
Constant: Never
0 = x(x + 2)
x = 0 and x = -2 Discontinuity at x = -1
2nd: Test Values
Interval
(-∞, -2)
(-2, -1)
(-1, 0)
(0 ,∞)
Test Value
x = -5
x = -3/2
x = -½
x=5
Sign of f’(x) f’(x) = (+)
f’(x) = (-)
f’(x) = (-)
Conclusion
Decreasing Decreasing Increasing
Increasing
f’(x) = (+)
Example 2: Find the open intervals on which the
following function is increasing or
decreasing.
x2
f ( x) 
x 1
Increasing: (-∞, -2)(0, ∞)
Decreasing: (-2, -1) (-1, 0)
Constant: Never
A function is STRICTLY MONOTONIC on an interval if
it is either increasing or decreasing on the entire
interval.
Example 3: Find the open intervals on which the function
f(x) = x3 increasing or decreasing.
1st: Find the critical numbers
f’(x) = 3x2
0 = 3x2
x=0
2nd: Test Values
Interval
(-∞, 0)
(0 ,∞)
Test Value
x = -5
x=5
Sign of f’(x) f’(x) = (+)
Conclusion
Increasing
f’(x) = (+)
Increasing
Increasing: (-∞, ∞ )
Decreasing: Never
Constant: Never
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(MUST USE CALCULUS!!!!)