1. Increasing, Decreasing, Constant
Objectives:
Be able to apply the first derivative test to find relative extrema of a function
Critical Vocabulary:
Increasing, Decreasing, Constant
Warm Up: Determine (using calculus) where each function is
increasing, decreasing, and constant.
1. f(x) = 2x3 - 3x2 - 36x + 14
2. f(x) = (x2 - 4)2/3 3.

2. This lesson is going to discuss the behavior of a critical number.
The previous lesson discussed the behavior of an interval (increasing, decreasing, and constant).
This lesson is going to discuss the behavior of a critical number.
If an interval changes from increasing to decreasing (or decreasing to increasing), then what is happening at the critical number that separates the intervals?
Interval
(-∞, 0)
(0, 1)
(1, ∞)
Test Value
x = -5
x = ½
x = 25
f’(x) = (+)
f’(x) = (-)
f’(x) = (+)
Sign of f’(x)
Conclusion
Increasing
Decreasing
Increasing 1
Relative Max
Relative
Min

3. 1. Use increasing/decreasing test to find the intervals.
Let c be a critical number of a function f that is continuous on the open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows:
If f’(x) changes from negative to positive at c, then f(c) is a relative minimum of f.
If f’(x) changes from positive to negative at c, then f(c) is a relative maximum of f.
If f’(x) does not change signs at c, then f(c) is neither a relative maximum nor relative minimum of f.
Summary:
1. Use increasing/decreasing test to find the intervals.
Use the First Derivative Test to determine if a critical number is a relative max or min.

4. Relative Min Relative Max
Directions: For the following exercises, find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing. Locate all the relative extrema. Use a graphing calculator to confirm your results.
Example 1: f(x) = 2x3 - 3x2 - 36x + 14
f’(x) = 6x2 - 6x - 36
0 = 6x2 - 6x - 36
0 = 6(x - 3)(x + 2)
x = 3 and x = -2
Interval
(-∞, -2)
(-2, 3)
(3, ∞)
Test Value
x = -4
x = 0
x = 4
Increasing:
(-∞, -2)
(3, ∞)
Decreasing:
(-2, 3)
Sign f’(x)
f’(x) = (+)
f’(x) = (-)
f’(x) = (+)
Constant:
Never
Conclusion
Increasing
Decreasing
Increasing -2 3
Relative Min
Relative Max
Relative Max: x = -2
Relative Min: x = 3

5. Relative Min Relative Min Relative Max
Directions: For the following exercises, find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing. Locate all the relative extrema. Use a graphing calculator to confirm your results.
Example 2: f(x) = (x2 - 4)2/3
4x = 0
x = 0
Undefined at x = 2 , x = -2
Interval
(-∞, -2)
(-2, 0)
(0, 2)
(2, ∞)
Increasing:
(-2, 0)
(2, ∞)
Test Value
x = -3
x = -1
x = 1
x = 3
Decreasing:
(-∞, -2)
(0, 2)
Sign f’(x)
f’(x) = (-)
f’(x) = (+)
f’(x) = (-)
f’(x) = (+)
Constant:
Never
Conclusion
Decreasing
Increasing
Decreasing
Increasing
Relative Max: x = 0 -2 2
Relative Min: x = 2
x = -2
Relative Min
Relative Min
Relative Max

6. Relative Min Relative Min
Directions: For the following exercises, find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing. Locate all the relative extrema. Use a graphing calculator to confirm your results.
Example 3:
2x4 – 2 = 0
x = 1 and x = -1
Discontinuity at x = 0
Interval
(-∞, -1)
(-1, 0)
(0, 1)
(1, ∞)
Increasing:
(-1, 0)
(1, ∞)
Test Value
x = -3
x = -½
x = ½
x = 3
Decreasing:
(-∞, -1)
(0, 1)
Sign f’(x)
f’(x) = (-)
f’(x) = (+)
f’(x) = (-)
f’(x) = (+)
Constant:
Never
Conclusion
Decreasing
Increasing
Decreasing
Increasing
Relative Max: None -1 1
Relative Min: x = 1
x = -1
Relative Min
Relative Min

7. Page #11-33 odd (skip 27), 55,
(MUST USE CALCULUS!!!!)