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Published byHilary Johnson Modified over 9 years ago
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Minimum and Maximum Values Section 4.1
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Definition of Extrema – Let be defined on a interval containing : i. is the minimum of on if ii. is the maximum of on if
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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.
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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. *
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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)
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*** 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!!***
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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.
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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.
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