5.1 analysis of function i
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This document discusses increasing and decreasing functions, concavity of functions, and finding intervals where functions are increasing, decreasing, concave up, or concave down. It provides examples of finding the intervals for the functions f(x)=x-4x^2+3 and f(x)=x-5x^4+9x showing the steps to determine where the functions are increasing or decreasing and where they are concave up or concave down. It also discusses inflection points and provides an example of finding intervals of increase, decrease, concavity and the inflection point for the function f(x)=x^3-3x^2+1.
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![How about an Example?
Find the intervals on which f (x) = x − 4x + 3 is increasing
and the intervals on which it is decreasing?
2
f '(x) = 2x − 4
f '(x) = 0
_ _ _ _ _ _
2x − 4 = 0
•
2
f '(x) < 0 ⇒ f (x)↓: ( − ∞,2]
f '(x) > 0 ⇒ f (x)↑: [ 2,+ ∞ )
+ + + + + +
x=2](https://image.slidesharecdn.com/5-131206204516-phpapp02/85/5-1-analysis-of-function-i-4-320.jpg)
![Let’s get all this together
pl e
m
xa
e
Considering f (x) = x 3 − 3x 2 +1 find the following : f (x)↑
3x ( x − 2 ) = 0
f '(x) = 3x 2 − 6x = 0
( − ∞,0] and [ 2,+ ∞ )
•
0
_ _ _
6 ( x −1) = 0
f '(x) = 6x − 6 = 0
_ _ _
•
2
+ + +
•
1
[ 0,2]
f (x)∪
( 1,+ ∞ )
f (x)∩
( −∞,1)
x
+ + +
f (x)↓
1
inf lection=
+ + +
f (1) = 13 − 3× 2 +1 = −1
1](https://image.slidesharecdn.com/5-131206204516-phpapp02/85/5-1-analysis-of-function-i-7-320.jpg)
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5.1 analysis of function i
- 1. Analysis of Functions: Increase, Decrease, and Concavity
- 4. How about an Example? Find the intervals on which f (x) = x − 4x + 3 is increasing and the intervals on which it is decreasing? 2 f '(x) = 2x − 4 f '(x) = 0 _ _ _ _ _ _ 2x − 4 = 0 • 2 f '(x) < 0 ⇒ f (x)↓: ( − ∞,2] f '(x) > 0 ⇒ f (x)↑: [ 2,+ ∞ ) + + + + + + x=2
- 5. How about another example? Find the intervals on which f (x) = x − 5x + 9x is concave up and the intervals on which it is concave down. 4 f '(x) = 4x 3 −15x 2 +18x f "(x) = 0 6 ( 2x 2 − 5x + 3) = 0 12x − 30x +18 = 0 • 1 - - 2 f "(x) = 12x 2 − 30x +18 2 + + + + + 3 - - - - •+ 3/2 3 f "(x) < 0 ⇒ f (x)∩ 1, ÷ 2 3 f "(x) > 0 ⇒ f (x)∪ ( − ∞,1) and ,+ ∞ ÷ 2 + + + + 6 ( 2x − 3) ( x −1) = 0
- 6. Inflection points Points of inflection occur at the points where: •the derivative equals zero •the derivative doesn’t exist •the function changes concavity Warning !!! Asymptotes frequently act as inflection points, so make sure you check concavity changes before making conclusion.
- 7. Let’s get all this together pl e m xa e Considering f (x) = x 3 − 3x 2 +1 find the following : f (x)↑ 3x ( x − 2 ) = 0 f '(x) = 3x 2 − 6x = 0 ( − ∞,0] and [ 2,+ ∞ ) • 0 _ _ _ 6 ( x −1) = 0 f '(x) = 6x − 6 = 0 _ _ _ • 2 + + + • 1 [ 0,2] f (x)∪ ( 1,+ ∞ ) f (x)∩ ( −∞,1) x + + + f (x)↓ 1 inf lection= + + + f (1) = 13 − 3× 2 +1 = −1 1
- 8. Find the intervals on which f(x) is increasing, decreasing, concave up, concave down and inflection point. 1. f (x) = xe −x 2. f (x) = x + 2sin x