Download Chapter 4 Section 1 - Columbus State University

Applications of Differentiation
Section 4.1
Maximum and
Minimum Values
Applications of Differentiation
 Maxima and Minima
 Applications of Maxima and Minima
Absolute Extrema
Let f be a function defined on a domain D
Absolute
Maximum
Absolute
Minimum
Absolute Extrema
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.
The number f (c) is called the absolute maximum
value of f in D
Absolute
Maximum
f (c )
c
Absolute Extrema
A 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 )
Absolute
Minimum
Generic Example
y
8
7
6
5
4
3
2
1
x
-3
-2
-1
1
-1
2
3
4
5
6
Generic Example
y
8
7
6
5
4
3
2
1
x
-3
-2
-1
1
-1
2
3
4
5
6
Generic Example
y
8
7
6
5
4
3
2
1
x
-3
-2
-1
1
-1
2
3
4
5
6
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
Fermat’s Theorem
If a function f has a local maximum or minimum at c,
and if f (c ) exists, then f (c)  0
Proof:
Assume f has a maximum
f (c  h )  f (c )
f (c)  lim
h 0
h
f (c  h )  f (c )
 lim
h 0
h
f (c  h )  f (c )
 lim
h 0
h
f ( c  h )  f (c )
f (c  h )  f (c )
 0 if h  0
h
f ( c  h )  f (c )
f (c)  lim
 0 if h  0
h 0
h
f ( c  h )  f (c )
f (c  h )  f (c )
 0 if h  0
h
f ( c  h )  f (c )
f (c)  lim
 0 if h  0
h 0
h
f (c)  0 and f (c)  0
Then f (c)  0
The Absolute Value of x.
f ( x)  x
f (0)  DNE
1 if x  0
f ( x)  
1 if x  0
Generic Example
y
8
f ( x)  0
7
6
5
4
3
f ( x)  DNE
2
1
x
-3
-2
-1
1
2
3
4
5
6
-1
The corresponding values of x are called
Critical Points of f
Critical Points of f
A critical number of a function f is a number c in
the domain of f such that
a. f (c)  0 (stationary point)
b. f (c ) does not exist (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
y
8
f ( x)  0
7
not a local extrema
6
5
4
3
f ( x)  DNE
2
not a local extrema
1
x
-3
-2
-1
1
2
3
4
5
6
-1
Two critical points of f that do
not correspond to local extrema
Example
Find all the critical numbers of
f ( x)  3 x 3  3 x .
x2 1
f ( x) 
3
x
3
 3x

2
Stationary points: x  1
Singular points: x  0,  3
Graph of f ( x)  x  3 x .
3
3
y
Local max. f (1)  3 2
2
1
x
-2
-1
1
2
3
-1
-2
-3
Local min. f (1)   3 2
Extreme Value Theorem
If 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
Attains max.
and min.
a
b
a
b
Attains min.
but not 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].
Example
 1 
Find the absolute extrema of f ( x)  x  3 x on   ,3 .
 2 
2
f ( x)  3x  6 x  3x( x  2)
3
2
Critical values of f inside the interval (-1/2,3) are x = 0, 2
Evaluate
f (0)  0
Absolute Max.
f (2)  4
Absolute Min.
7
 1
f    
8
 2
f  3  0
Absolute Max.
Example
 1 
Find the absolute extrema of f ( x)  x  3 x on   ,3 .
 2 
3
2
Critical values of f inside the interval (-1/2,3) are x = 0, 2
Absolute Max.
-2
-1
1
2
3
4
5
6
Absolute Min.
-5
Example
 1 
Find the absolute extrema of f ( x)  x  3x on   ,1 .
 2 
2
f ( x)  3x  6 x  3x( x  2)
3
2
Critical values of f inside the interval (-1/2,1) is x = 0 only
Evaluate
f (0)  0
Absolute Max.
7
 1
f    
8
 2
f 1  2
Absolute Min.
Example
 1 
Find the absolute extrema of f ( x)  x  3x on   ,1 .
 2 
2
f ( x)  3x  6 x  3x( x  2)
3
2
Critical values of f inside the interval (-1/2,1) is x = 0 only
Absolute Max.
-2
-1
1
2
3
4
5
6
Absolute Min.
-5