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

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. *

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)

*** 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 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.

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.