November 08, 2010
Section 3.1 Extrema on an Interval
Definition 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 .
A min or max of f on I is called the extremum .
Extreme Value Theorem
If f is continuous on [ a , b ], then f has both a min and max on [ a , b ].
Definition of Relative Extrema
1. If there is an open interval containing c on which
f ( c ) is a maximum, then f ( c ) is called a relative
maximum of f , or we say " f has a relative max at
2. If there is an open interval containing c on which
f ( c ) is a minimum, then f ( c ) is called a relative
minimum of f , or we say " f has a relative min at
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November 08, 2010
Definition of a Critical Number
Let f be defined at c . If f ' ( c ) = 0 or if f is not differentiable at c , then c is a critical number of f .
Theorem 3.2
If f has a relative min or max at x = c , then c is a critical number of f .
Guidelines to find the extrema of a continuous function f on [ a, b ]
1. Find the critical numbers of f in ( a , b ),
2. Evaluate f at each critical number in ( a , b ),
3. Evaluate f at each endpoint of [ a , b ],
4. The least of these values in the min. The greatest is the max.
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Find the critical numbers for
November 08, 2010
#40 If f ( x ) = √ 4 − x 2 locate the absolute
extrema over the following intervals.
[-2, 2] [-2, 0) (-2, 2) [1, 2)
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November 08, 2010
Locate the absolute extrema for f over the given interval.
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November 08, 2010
#42 Sketch the graph of f and locate the
absolute extrema over the interval [1, 5].
f ( x ) =



2 − x 2 , 1 ≤ x < 3
2 − 3 x , 3 ≤ x ≤ 5
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Find the critical numbers of over the interval 0 ≤ π < 2 π .
November 08, 2010
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November 08, 2010
Locate the absolute extrema of the function on the interval [-5, -1].
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