Review 3.1-3.3 - Increasing or Decreasing - Relative Extrema - Absolute Extrema - Concavity
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Transcript of Review 3.1-3.3 - Increasing or Decreasing - Relative Extrema - Absolute Extrema - Concavity
Review 3.1-3.3
- Increasing or Decreasing- Relative Extrema- Absolute Extrema
- Concavity
-3 -2 -1 1 2 3 4 5 6
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Increasing/Decreasing/Constant
-3 -2 -1 1 2 3 4 5 6
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Increasing/Decreasing/Constant
.,on increasing is then
,, intervalan in of each valuefor 0 Ifbaf
baxxf
.,on decreasing is then
,, intervalan in of each valuefor 0 Ifbaf
baxxf
.,on constant is then
,, intervalan in of each valuefor 0 Ifbaf
baxxf
Increasing/Decreasing/Constant
Generic Example
-3 -2 -1 1 2 3 4 5 6
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( ) 0f x
( )f x DNE
The corresponding values of x are called Critical Points of f
Critical Points of f
a. ( ) 0f c
A critical number of a function f is a number c in the domain of f such that
b. ( ) does not existf c
ExampleFind all the critical numbers of
When set = 0Excluded values
Determine where the function is increasing or decreasing:
-1 0
Plug into the derivate around each critical #
- - - - - + + + + + +
Increasing: (-1, ∞)Decreasing: (-∞, -1)
ExampleFind all the critical numbers of
When set = 0Excluded values
Determine where the function is increasing or decreasing:
-1 0 1
Plug into the derivate around each critical #
Not in Domain + + + - - - - - Not in Domain
Increasing: (-1, 0)Decreasing: (0, 1)
Example
3 3( ) 3 .f x x x
2
233
1( )3
xf xx x
Find all the critical numbers of
0, 3x
When set = 0 1x Excluded values
Determine where the function is increasing or decreasing:
-1 0 1
Plug into the derivate around each critical #
+ + + + + + - - - - - - - - - - + + + + + +
Increasing: (- ∞, -1) U (1, ∞)Decreasing: (-1, 1)
Graph of 3 3( ) 3 . f x x x
-2 -1 1 2 3
-3
-2
-1
1
2
x
y
Local max. 3( 1) 2f
Local min. 3(1) 2f
If the price of a certain item is p(x) and the total cost to produce x units is C(x), at what production levels is profit increasing and decreasing?
Now find P’(x)
Now test around 18, -2
Increasing: (0, 18)Decreasing: (18, ∞)
Relative ExtremaA function f has a relative (local) maximum at x = c if there exists an open interval (r, s) containing c such that f (x) = f (c)
Relative Maxima
Relative ExtremaA function f has a relative (local) minimum at x = c if there exists an open interval (r, s) containing c such that f (c) = f (x)
Relative Minima
The First Derivative Test
left right
f (c) is a relative maximum
f (c) is a relative minimum
No change No relative extremum
Determine the sign of the derivative of f to the left and right of the critical point.
conclusion
The First Derivative Test.16)( 23 xxxf
2( ) 3 12 0f x x x
Find all the relative extrema of
0)4(3 xx4,0x
0 4
+ 0 - 0 +
Relative max. f (0) = 1
Relative min. f (4) = -31
f
f
Excluded Values: None
-2 -1 1 2 3 4 5 6 7 8 9 10
-35
-30
-25
-20
-15
-10
-5
5
x
y
(4,f(4)=-31)
(0,f(0)=1)
The First Derivative Test
The First Derivative TestFind all the relative extrema of
Excluded Values: None
-1 0 1
+ + + - - - - - - - - - - + + +
Rel. Min. (1, -2)Rel. Max. (-1, 2)
-1 0 1
+ ND + 0 - ND - 0 + ND +
Relative max. Relative min.
f
f
0, 3x
1x
Exclude Values:3(1) 2f
3
3( 1) 2f
3
3 3( ) 3 .f x x x
Example from before:
2
233
1( )3
xf xx x
-2 -1 1 2 3
-3
-2
-1
1
2
x
y
Rel. max. 3( 1) 2f
Rel. min. 3(1) 2f
Graph of 3 3( ) 3 . f x x x
Absolute Extrema
Absolute Minimum
Let f be a function defined on a domain D
Absolute Maximum
The number f (c) is called the absolute maximum value of f in D
A function f has an absolute (global) maximum at x = c if f (x) = f (c) for all x in the domain D of f.
Absolute Maximum
Absolute Extrema
c
( )f c
Absolute Minimum
Absolute ExtremaA function f has an absolute (global) minimum at x = c if f (c) = f (x) for all x in the domain D of f.
The number f (c) is called the absolute minimum value of f in D
c
( )f c
Finding absolute extrema on [a , b]
1. Find all critical numbers for f (x) in (a , b).2. Evaluate f (x) for all critical numbers in (a , b).3. Evaluate f (x) for the endpoints a and b of the
interval [a , b]. 4. The largest value found in steps 2 and 3 is the
absolute maximum for f on the interval [a , b], and the smallest value found is the absolute minimum for f on [a , b].
ExampleFind the absolute extrema of 3 2 1( ) 3 on ,3 .
2f x x x 2( ) 3 6 3 ( 2)f x x x x x
Critical values of f inside the interval (-1/2,3) are x = 0, 2
(0) 0(2) 4
1 72 8
3 0
ff
f
f
Absolute Max.
Absolute Min.Evaluate
Absolute Max.
ExampleFind the absolute extrema of 3 2 1( ) 3 on ,3 .
2f x x x
Critical values of f inside the interval (-1/2,3) are x = 0, 2
Absolute Min.
Absolute Max.
-2 -1 1 2 3 4 5 6
-5
ExampleFind the absolute extrema of 3 2 1( ) 3 on ,1 .
2f x x x 2( ) 3 6 3 ( 2)f x x x x x
Critical values of f inside the interval (-1/2,1) is x = 0 only
(0) 01 72 8
1 2
f
f
f
Absolute Min.
Absolute Max.
Evaluate
-2 -1 1 2 3 4 5 6
-5
ExampleFind the absolute extrema of 3 2 1( ) 3 on ,1 .
2f x x x 2( ) 3 6 3 ( 2)f x x x x x
Critical values of f inside the interval (-1/2,1) is x = 0 only
Absolute Min.
Absolute Max.
ConcavityLet f be a differentiable function on (a, b).
1. f is concave upward on (a, b) if f ' is increasing on aa(a, b). That is f ''(x) > 0 for each value of x in (a, b).
concave upward concave downward
2. f is concave downward on (a, b) if f ' is decreasing on (a, b). That is f ''(x) < 0 for each value of x in (a, b).
Inflection PointA point on the graph of f at which f is continuous and concavity changes is called an inflection point.
To search for inflection points, find any point, c in the domain where f ''(x) = 0 or f ''(x) is undefined.
If f '' changes sign from the left to the right of c, then (c, f (c)) is an inflection point of f.
Example: Inflection Points
.16)( 23 xxxf2( ) 3 12f x x x
Find all inflection points of
( ) 6 12f x x Possible inflection points are solutions of a) ( ) 0 b) ( ) 6 12 0 no solutions 2
f x f x DNEx
x
2
- 0 +
Inflection point at x 2
f
f
-2 -1 1 2 3 4 5 6 7 8 9 10
-35
-30
-25
-20
-15
-10
-5
5
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y
(4,f(4)=-31)
(0,f(0)=1)