Math 151 Section 5.2 Local and Absolute Max and Min
Absolute Max and Min:
For f(x) defined on a set D, and c in D
a) f ( c ) is an absolute maximum
of f on D if
f ( x )  f ( c ) for all x in D.
b)
f ( c ) is an absolute minimum
of f on D if
f ( x )  f ( c ) for all x in D.
Local Max and Min:
For f(x) defined on a set D, and c in D
a) f ( c ) is a local maximum
of f on D if
f ( x )  f ( c ) for all x in some interval ( c  r , c  r ).
b)
f ( c ) is a local minimum
of f on D if
f ( x )  f ( c ) for all x in some interval ( c  r , c  r ).
How to find local max and min using the first derivative: Find the derivative, make a sign chart or view
the graph of the derivative. If f (c ) is defined and if f ' ( x ) changes from positive to negative as x
crosses over c , then f (c ) is a local maximum.
If f (c ) is defined and if f ' ( x ) changes from negative to positive as x crosses over c , then f (c ) is
a local minimum.
Theorem: If f ( x ) has either a local max or min at c , then f ' ( c )  0 or f ' ( c ) does not exist.
Examples: Find any local max and min of the function.
1.
f (x)  x  3x  9 x  7
2.
f ( x )  ( x  1) ( x  2 )
3.
3
2
3
f ( x) 
x
2
x2
Finding Absolute Max and Min:
Extreme Value Theorem: If f ( x ) is continuous on the closed, bounded interval, [ a , b ] ,
then f ( x ) has both an absolute max and an absolute min on [ a , b ] and these must occur at
local extreme points or at the endpoints of the interval.
To find the absolute max and min of a continuous function, find all points in the given interval where
f ' ( c )  0 or f ' ( c ) does not exist. List these with the endpoints of the given interval and compare
the function values.
Examples: Find the absolute max and min of the function on the given interval.
1.
f ( x )  2 x  3 x  12 x
[  2 ,0 ]
2.
f ( x )  2 x  3 x  12 x
[  2 ,3 ]
3
3
2
2