Maximum and Minimum Values:
Def:
1. A function f has an absolute maximum (global maximum) at
c
if( ) ( ) ( )
f c ≥ f x for all x in D f
The number f c( ) is called the maximum value of f on D f( )
2. A function f has an absolute minimum (global minimum) at
c
if( ) ( ) ( )
f c ≤ f x for all x in D f
The number f c( ) is called the minimum value of f on D f( )
3. The maximum and the minimum
values of f are called the extreme values of f
Def:
1. A function f has a local maximum (Relative maximum) at
c
if( ) ( )
f c ≥ f x When x is near
c
2. A function f has a local minimum (Relative minimum) at
c
if( ) ( )
f c ≤ f x When x is near
c
The Extreme Value Theorem:
If f is continuous on a closed interval [ , ]a b , then f attains an absolute maximum value
( )
f c and an absolute minimum value f d( ) at some numbers
c
and d in [ , ]a b .Fermat’s Theorem:
If f has a local maximum or minimum at
c
, and if f c′( ) exist, then f c′( )=0Def: A critical number of a function f is a number
c
in the domain of f such thateither f c′( ) 0= or f c′( ) does not exist
EX: find the critical numbers of the function
1.
3
( )
5(4 ) f x = x − x
2. g( )θ = +θ sinθ3.
f t ( ) 2 = t
3+ 3 t
2+ + 6 t 5
4. ( ) 21 g x x
= x
+ 5. h x( ) = x xln
To find the absolute maximum and
minimum values of a continuous function f on the a closed interval [ , ]a b
1. find the critical numbers of f in ( , )a b 2. find the values of f at the numbers in
step 1
3. find the values of f at the endpoints of the interval
4. the largest of the values in step 2 and step3 is the absolute maximum; the smallest of those values is the absolute minimum.
Ex: find the absolute maximum and minimum of the function on the given interval
1.
f x ( ) = − x
33 x
2+ 2
[1,4]2.
f x ( ) 3 = x
3− 3 x + 1
[0,3]
