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Extrema On the Interval

Extrema On the Interval - Theorem 3.2 Relative...

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Extrema On the Extrema On the Interval Interval Section 3-1
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Defn. of Extrema Defn. of Extrema Let f be defined on an interval I containing c. (1) f(c) is the minimum of f on I if f(c) < f(x) for all x in I. (2) f(c) is the maximum of f on I if f(c) > f(x) for all x in I.
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Theorem 3.1 Theorem 3.1 If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval. Extrema can occur at endpoints or critical pts.
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Relative extrema & critical #s Relative extrema & critical #s If hill or valley is rounded, the graph has a horizontal tangent line at the high or low point (f’(c) = 0) If the hill or valley is sharp & peaked, the graph represents a function that is not differentiable at the high/low points.
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Theorem 3.2
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Unformatted text preview: Theorem 3.2 Relative minimums/maximums occur only at critical numbers. At relative min/maxs, the derivative is either 0 or undefined. Critical Numbers Critical Numbers Let f be defined at c. If f ’(c) = 0 or if f ‘ is undefined at c, then c is a critical number of c. Finding Extrema On a Closed Interval Finding Extrema On a Closed Interval (1) Find the critical #s of f in (a , b). (2) Evaluate f at each critical # in (a , b). (3) Evaluate f at each endpoint of [a , b]. (4) The least of these values is the minimum. The greatest is the maximum. Joke Time Joke Time What kind of undergarment does Little Mermaid wear? an Alge-bra Why did the man sleep with a ruler? to see how long he could sleep...
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