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3.1 Extrema On An Interval

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After this lesson, you should be able to: Understand the definition of extrema of a function on an interval Understand the definition of relative extrema of a function on an open interval Find extrema on a closed interval

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Definition

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When the just word minimum or maximum is used, we assume its an absolute min or absolute max. Extrema Minimum and maximum values on an interval are called extremes, or extrema on an interval. The minimum value of the function on an interval is considered the absolute minimum on the interval. The maximum value of the function on an interval is considered the absolute maximum on the interval.

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OPEN intervals – Do the following have extrema? On an open interval, the max. or the min. may or may not exist even if the function is continuous on this interval.

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CLOSED intervals – Do the following have extrema? On a closed interval, both max. and min. exist if the function is continuous on this interval.

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The Extreme Value Theorem (EVT) 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. In other words, if f is continuous on a closed interval, f must have a min and a max value. Max-Min f is continuous on [ a, b ] a b

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Example Example 1 Let f ( x ) = x 2 – 5 x – 6 on the closed interval [–1, 6], find the extreme values.

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Example Example 2 Let f ( x ) = x x 2 – x – 2 on the closed interval [–3, 1], find the extreme values.

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Example Example 3 Let f ( x ) = x x 2 – x – 2 on the closed interval [–3, 2], find the extreme values. The (absolute)max and (absolute)min of f on [ a, b ] occur either at an endpoint of [ a, b ] or at a point in ( a, b ).

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Relative Extrema and Critical Numbers (AP may use Local Extrema)

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1.If there is an open interval containing c on which f ( c ) is a maximum, then f ( c ) is a local maximum of f. 2.If there is an open interval containing c on which f ( c ) is a minimum, then f ( c ) is a local minimum of f. When you look at the entire graph (domain), there may be no absolute extrema, but there could be many relative extrema. What is the slope at each extreme value????

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Definition of a Critical Number and Figure 3.4

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Critical Numbers c is a critical number for f iff: f ( c ) is defined ( c is in the domain of f ) f ( c ) = 0 or f ( c ) = does not exist Theorem 3.2 If f has a relative max. or relative min, at x = c, then c must be a critical number for f. The (absolute)max and (absolute)min of f on [ a, b ] occur either at an endpoint of [ a, b ] or at a critical number in ( a, b ). Explain this statement. So…. Relative extrema can only occur at critical values, but not all critical values are extrema. Explain this statement.

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Make sure f is continuous on [ a, b ]. 1.Find the critical numbers of f ( x ) in ( a, b ). This is where the derivative = 0 or is undefined. 2.Evaluate f ( x ) at each critical numbers in ( a, b ). 3.Evaluate f ( x ) at each endpoint in [ a, b ]. 4.The least of these values (outputs) is the minimum. The greatest is the maximum. **Make sure you give the y-value this is the extreme value!** Guidelines

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Critical Numbers 1. Make sure f is continuous on [ a, b ]. 2. Find all critical numbers c 1, c 2, c 3 …c n of f which are in ( a, b ) where f ( x ) = 0 or f ( x ) is undefined. 3. Evaluate f ( a ), f ( b ), f ( c 1 ), f ( c 2 ), … f ( c n ). 4. The largest and smallest values in part 2 are the max and min of f on [ a, b ]. To find the max and min of f on [ a, b ]:

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Example Example 4 Find all critical numbers Domain: (–, + ) Critical number: x = 2 and x = 5

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Example Example 5 Find all critical numbers. Domain: x 1, x R Critical number: x = 1, x = 0, and x = 2

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Example Example 6 Find all critical numbers. Domain: (–, + ) Critical number: x = –4 f (–4 )

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Example x Left EndpointCritical Number Right Endpoint f (x) f (– 3 ) = 20 f (– 2 ) = 30 f ( 4 ) = – 78 f ( 5 ) = – 68 Example 7 Find the max and min of f on the interval [–3, 5]. Domain: (–, + ) Critical number: x = –2 and x = 4 minimummaximum Graph is not in scale

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Practice Of x Left EndpointCritical Number Right Endpoint f (x) f (– 1 ) = 7 f ( 0 ) = 0 f ( 1 ) = – 1 f ( 2 ) =16 Example 7 Find the extrema of f on the interval [–1, 2]. Domain: (–, + ) Critical number: x = 0 and x = 1 minimummaximum

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Example Example 8 Find the extrema of f on the interval [–1, 3]. Critical number: x = 0 and x = 1 x Left EndpointCritical Number Right Endpoint f (x) f (– 1 ) = – 5 f ( 0 ) = 0 f ( 1 ) = – 1 f (3) = minimummaximum f (0) does not exist

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Practice Of Example 8 Find the extrema of f on the interval [0, 2 ]. Critical number: x = /2, x = 3 /2, x = 7 /6, x = 11 /6 x Left Endpoint Critical Number Right Endpoint f (x) f (0) = – 1 f ( /2 ) = 3 f (7 /6 ) = – 3/2 f (3 /2 ) = – 1 f (11 /6 ) = – 3/2 f (2 ) = – 1 minmaxmin

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Summary Open vs. Closed Intervals 1. An open interval MAY have extrema 2. A closed interval on a continuous curve will ALWAYS have a minimum and a maximum value. 3. The min & max may be the same value How?

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Homework Section 3.1 page 165 #1-23 odd, 55

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