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Chapter 5Applications of the
Derivative
Sections 5.1, 5.2, 5.3, and 5.4
Applications of the Derivative
Maxima and Minima
Applications of Maxima and Minima
The Second Derivative - Analyzing Graphs
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
-3 -2 -1 1 2 3 4 5 6
-1
1
2
3
4
5
6
7
8
x
y
Generic Example
-3 -2 -1 1 2 3 4 5 6
-1
1
2
3
4
5
6
7
8
x
y
Generic Example
-3 -2 -1 1 2 3 4 5 6
-1
1
2
3
4
5
6
7
8
x
y
Generic Example
Relative Extrema
A 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) for all r x s.
Relative Maxima
Relative Extrema
A 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) for all r x s.
Relative Minima
Generic Example
-3 -2 -1 1 2 3 4 5 6
-1
1
2
3
4
5
6
7
8
x
y
( ) 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(stationary point)
(singular point)
Candidates for Relative Extrema
1.Stationary points: any x such that x is in the domain of f and f ' (x) 0.
2.Singular points: any x such that x is in the domain of f and f ' (x) undefined
3. Remark: notice that not every critical number correspond to a local maximum or local minimum. We use “local extrema” to refer to either a max or a min.
Fermat’s Theorem
If a function f has a local maximum or minimum at c, then c is a critical number of f
Notice that the theorem does not say that at every critical number the function has a local maximum or local minimum
Generic Example
-3 -2 -1 1 2 3 4 5 6
-1
1
2
3
4
5
6
7
8
x
y
( )
not a local extrema
f x DNE
Two critical points of f that donot correspond to local extrema
( ) 0
not a local extrema
f x
Example
3 3( ) 3 .f x x x
2
233
1( )
3
xf x
x x
Find all the critical numbers of
0, 3x Stationary points: 1x Singular points:
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
Extreme Value TheoremIf a function f is continuous on a closed interval [a, b], then f attains an absolute maximum and absolute minimum on [a, b]. Each extremum occurs at a critical number or at an endpoint.
a b a ba b
Attains max. and min.
Attains min. but no max.
No min. and no max.
Open Interval Not continuous
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 .2
f 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 7
2 8
3 0
f
f
f
f
Absolute Max.
Absolute Min.Evaluate
Absolute Max.
ExampleFind the absolute extrema of 3 2 1
( ) 3 on ,3 .2
f 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 .2
f 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) 0
1 7
2 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 .2
f 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.
-3 -2 -1 1 2 3 4 5 6
-1
1
2
3
4
5
6
7
8
x
y
Increasing/Decreasing/Constant
-3 -2 -1 1 2 3 4 5 6
-1
1
2
3
4
5
6
7
8
x
y
Increasing/Decreasing/Constant
.,on increasing is then
,, intervalan in of each valuefor 0 If
baf
baxxf
.,on decreasing is then
,, intervalan in of each valuefor 0 If
baf
baxxf
.,on constant is then
,, intervalan in of each valuefor 0 If
baf
baxxf
Increasing/Decreasing/Constant
The First Derivative Test
Generic Example
-3 -2 -1 1 2 3 4 5 6
-1
1
2
3
4
5
6
7
8
x
y
c
( ) 0
to the right of
f x
c
A similar ObservationApplies at aLocal Max.
( ) 0
to the left of
f x
c
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
Relative Extrema
3 2( ) 6 1f x x x
2( ) 3 12 0f x x x
Example: Find all the relative extrema of
4,0xStationary points:
Singular points: None
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
The First Derivative Test
-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)
-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
Another Example
3 3( ) 3 .f x x x
2
233
1( )
3
xf x
x x
Find all the relative extrema of
0, 3x
Stationary points: 1x
Singular points:
-1 0 1
+ ND + 0 - ND - 0 + ND +
Relative max. Relative min.
f
f
0, 3x Stationary points: 1x Singular points:
3(1) 2f
3
3( 1) 2f
3
-2 -1 1 2 3
-3
-2
-1
1
2
x
y
Local max. 3( 1) 2f
Local min. 3(1) 2f
Graph of 3 3( ) 3 . f x x x
Domain Not a Closed IntervalExample: Find the absolute extrema of
1( ) on 3, .
2f x
x
Notice that the interval is not closed. Look graphically:
Absolute Max.
(3, 1)
Optimization Problems1. Identify the unknown(s). Draw and label a diagram as
needed.
2. Identify the objective function. The quantity to be minimized or maximized.
3. Identify the constraints.
4. State the optimization problem.
5. Eliminate extra variables.
6. Find the absolute maximum (minimum) of the objective function.
Optimization - ExamplesAn open box is formed by cutting identical squares from each corner of a 4 in. by 4 in. sheet of paper. Find the dimensions of the box that will yield the maximum volume.
xx
x
4 – 2x
4 – 2xx
(4 2 )(4 2 ) ; in 0,2V lwh x x x x
2( ) 16 32 12V x x x 4(2 3 )(2 )x x
Critical points: 2
2, both in [0,2]3
x
3
(2) 0
(0) 0
24.74 in
3
V
V
V
The dimensions are 8/3 in. by 8/3 in. by 2/3 in. giving a maximum box volume of V 4.74 in3.
2 316 16 4V x x x x
An metal can with volume 60 in3 is to be constructed in the shape of a right circular cylinder. If the cost of the material for the side is $0.05/in.2 and the cost of the material for the top and bottom is $0.03/in.2 Find the dimensions of the can that will minimize the cost.
2 60V r h 2(0.03)(2) (0.05)2C r rh
top and bottom
sidecost
Optimization - Examples
2 60V r h
22
60(0.03)(2) (0.05)2r r
r
2
60h
rSo
2(0.03)(2) (0.05)2C r rh
2 60.06 r
r
2
60.12C r
r
2
60 gives 0.12C r
r
36
2.52 in. which yields 3.02 in.0.12
r h
Sub. in for h
So with a radius ≈ 2.52 in. and height ≈ 3.02 in. the cost is minimized at ≈ $3.58.
Graph of cost function to verify absolute minimum:
2.5
Second Derivative
If ( ) is a function of , then the function
( ) denotes the first derivative of ( ).
Now, the derivative of ( ) is denoted by
( ) and called the second derivative of
the function
y f x x
y f x f x
y f x
y f x
y
( ).f x2
2Notation: ( ) is also denoted by
d ff x
dx
2 3Given ( ) 130 15 8 find (2).s t t t s
2( ) 30 24s t t t
( ) 30 48s t t
then, (2) 30 48(2) 66s
Second Derivative - Example
In both cases is increasing. However, in the first case
curves down and in the second case curves up.
f
f f
Second Derivative
( ) is
( ) i
so,
s
( ) 0
f x
f x
f x
( ) is
( ) i
so,
s
( ) 0
f x
f x
f x
Second Derivative
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 DNE
x
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
x
y
(4,f(4)=-31)
(0,f(0)=1)
The Point of Diminishing Returns
( ) 120 6S t t
If the function represents the total sales of a particular object, t months after being introduced, find the point of diminishing returns.
2 3( ) 100 60S t t t
2( ) 120 3S t t t
S concave up on
S concave down on 0,20 20,
The point of diminishing returns is at 20 months (the rate at which units are sold starts to drop).
10 20 30 40
5000
10000
15000
20000
25000
30000
t
S(t)
S concave up on
S concave down on
0,20
20,
Inflection point
The Point of Diminishing Returns