Transcript Presentation Version - Parkway C-2

Objectives:
1. 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
x4 1
3. f ( x) 
x2
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
Test Value
(-∞, 0)
(0, 1)
x = -5
x=½
Sign of f’(x) f’(x) = (+)
Conclusion
x = 25
f’(x) = (-)
Increasing
f’(x) = (+)
Decreasing
0
Relative
Max
(1, ∞)
Increasing
1
Relative
Min
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:
1. If f’(x) changes from negative to positive at c, then f(c) is
a relative minimum of f.
2. If f’(x) changes from positive to negative at c, then f(c) is
a relative maximum of f.
3. 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.
2. Use the First Derivative Test to determine if a
critical number is a relative max or 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 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
Test Value
(-∞, -2)
(-2, 3)
x = -4
x=0
(3, ∞)
x=4
Sign f’(x) f’(x) = (+) f’(x) = (-)
f’(x) = (+)
Conclusion Increasing Decreasing Increasing
-2
Relative
Max
3
Increasing: (-∞, -2) (3, ∞)
Decreasing: (-2, 3)
Constant: Never
Relative
Min
Relative Max: x = -2
Relative Min: x = 3
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
f ' ( x) 
33 x 2  4
4x = 0
x = 0 Undefined at x = 2 , x = -2
Interval
(-∞, -2)
(-2, 0)
(0, 2)
(2, ∞)
x = -3
x = -1
x=1
x=3
Test Value
Sign f’(x)
f’(x) = (-) f’(x) = (+)
f’(x) = (-)
f’(x) = (+)
Conclusion Decreasing Increasing Decreasing Increasing
Relative Min
-2
0
Relative Max
2
Increasing: (-2, 0) (2, ∞)
Decreasing: (-∞, -2) (0, 2)
Constant: Never
Relative Max: x = 0
Relative Min: x = 2
x = -2
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.
x4 1
Example 3: f ( x) 
x2
2x4  2
f ' ( x) 
x3
2x4 – 2 = 0
x = 1 and x = -1 Discontinuity at x = 0
Interval
(-∞, -1)
Test Value
Sign f’(x)
(-1, 0)
x = -3
(0, 1)
x = -½
(1, ∞)
x=½
f’(x) = (-) f’(x) = (+)
x=3
f’(x) = (-)
f’(x) = (+)
Conclusion Decreasing Increasing Decreasing Increasing
Relative Min
-1
0
1
Increasing: (-1, 0) (1, ∞)
Decreasing: (-∞, -1)(0, 1)
Constant: Never
Relative Max: None
Relative Min: x = 1
x = -1
Relative Min
Page 334-335 #11-33 odd (skip 27), 55,
(MUST USE CALCULUS!!!!)