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2224-Sec12_7-HWT - Mat h 22 24 Multiva ri a ble C alc Se c...

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Math 2224 Multivariable Calc–Sec. 12.7: Extreme Values and Saddle Points I. Review from 1205 A. Definitions 1. Local Extreme Values (Relative) a. A function f has a local maximum value at an interior point c of its domain if f x ( ) ! f c ( ) for all x in an open interval containing c . b. A function f has a local minimum value at an interior point c of its domain if f x ( ) ! f c ( ) for all x in an open interval containing c . 2. Critical Number a. A critical number of a function f is a number c in the domain of f such that ! f ( c ) is zero or ! f ( c ) does not exist (undefined). (A stationary point exists where ! f ( x ) = 0 and a singular point exists where ! f ( x ) is undefined.) b. If f has a local maximum or minimum at c , then c is a critical number of f . 3. A point where the graph of a function has a tangent line and where the concavity changes is called a point of inflection . B. The First Derivative Test for Local Extrema Let f be a continuous function on [ a,b ] and c be a critical number in [ a,b ]. 1. If ! f ( x ) " 0 on ( a,c ) and ! f ( x ) " 0 on ( c,b ), then f has a local maximum of f(c) at x=c . 2. If ! f ( x ) " 0 on ( a,c ) and ! f ( x ) " 0 on ( c,b ), then f has a local minimum of f(c) at x=c . 3. If ! f ( x ) does not change signs at x=c , then f has no local extrema at x=c .
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