Minimum and Maximum Values

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.