# 4.1

```4.1 MAXIMUM AND MINIMUM VALUES
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]
Definition: Let c be a number in the domain D of a function f. Then f(c) is
the:
ABSOLUTE MAXIMUM value of f on D
if f (c)  f ( x) for all x in D
ABSOLUTE MINIMUM value of f on D
if f (c)  f ( x) for all x in D
Definition: The number f(c) is :
LOCAL MAXIMUM value of f if f (c)  f ( x) when x is near c.
LOCAL MINIMUM value of f if f (c)  f ( x) when x is near c.
Critical Numbers are the x values where
Find Critical Numbers
f ' ( x)  0
or is undefined
Find the critical numbers of the function
f ( x)  x 3  6 x 2  15 x
f ( )  4  tan 
f ( x)  x 4  x 3  x 2  1
f ( x)  x 
1
x
f ( x )  ( x 2  1)3
f ( x) 
x2
x2  9
Find the absolute and local maximum and minimum values of
f ( x)  x 3  6 x 2  5 on [3,5]
f ( x)  x  ln x on [ 12 ,2]
f ( x)  2 cos x  sin 2 x on [0,  ]
```