Minimum and Maximum ValuesSection 4.1

Definition of Extrema – Let be defined on a interval containing :

i. is the minimum of on if

ii. is the maximum of on if

f

f Ic

( )f c f I( ) ( )f c f x x I

( )f c f I( ) ( )f c f x x I

Extreme Values (extrema) – minimum and maximum of a function on an interval

{can be an interior point or an endpoint}

Referred to as absolute minimum, absolute maximum and endpoint extrema.

Extreme Value Theorem: {EVT}If is continuous on a closed

interval then has both a minimum

and a maximum on the interval.

* This theorem tells us only of the existence of a maximum or minimum value – it does not tell us how to find it. *

f

,a b f

Definition of a Relative Extrema:i. If there is an open interval on

which is a maximum, then

is called a relative maximum of . (hill)

ii. If there is an open interval on which

is a maximum, then is called a relative minimum of . (valley)

( )f c ( )f c

f

( )f c ( )f c

f

*** Remember hills and valleys that are smooth and rounded have horizontal tangent lines. Hills and valleys that are sharp and peaked are not differentiable at that point!!***

Definition of a Critical NumberIf is defined at , then is called

a critical number of , if or if .

**Relative Extrema occur only at Critical Numbers!!**

If f has a relative minimum or relative maximum at x=c , then c is a critical number of f.

f

fc c

'( ) 0f c

'( )f c und

Guidelines for finding absolute extremai. Find the critical numbers of .

ii. Evaluate at each critical number in .

iii. Evaluate at each endpoint .

iv. The least of these y values is the minimum and the greatest y value is the maximum.

1' 0 '0

f y or y

f

,a bf ,a b