Minimum and Maximum Values - PowerPoint Presentation

Slide1

Minimum and Maximum Values

Section 4.1Slide2

Definition of

Extrema

–

Let be defined on a interval containing : i. is the minimum of on if ii. is the maximum of on if Slide3

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.Slide4

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. * Slide5

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)Slide6

*** 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!!***Slide7

Definition of a Critical Number

If 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.Slide8

Guidelines for finding

absolute

extrema

i. 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.