1 / 8

Minimum and Maximum Values

Minimum and Maximum Values. Section 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 .

rafi
Télécharger la présentation

Minimum and Maximum Values

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Minimum and Maximum Values Section 4.1

  2. Definition of Extrema – Let be defined on a interval containing : i. is the minimum of on if ii. is the maximum of on if

  3. 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.

  4. 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. *

  5. 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)

  6. *** Remember hills and valleys that are smooth and rounded have horizontal tangent lines. Hills and valleys that are sharp and peaked are notdifferentiable at that point!!***

  7. 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.

  8. 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.

More Related