Extreme Values and Saddle Points

Extreme Values and Saddle Points - [ a, b ] 1. Find...

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12.7 Extreme Values and Saddle Points In R 2 : Tests 1. 1st Derivative Test 2. 2nd Derivative Test In R 3 : Local extrema occur at critical points . Defn. Critical Point A critical point is an interior point where Note 1
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Defn. Saddle Point A point ( a, b, f ( a, b )) is a saddle point if 1. 2. In summary, there are 3 kinds of local extrema: 1. 2. 3. 2
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The Second Derivative Test for Local Extrema Let z = f ( x, y ) and suppose f ( a, b ) = ~ 0. Define the ”discriminant of f ” as follows: Then, 1. 2. 3. 4. Example 1 Find the local extrema of f ( x, y ) = x 4 + y 3 + 32 x - 9 y . 3
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Example 2 f ( x, y ) = xy 2 - 6 x 2 - 3 y 2 Do. Find all local extrema of f ( x, y ) = 6 x 2 - 2 x 3 + 3 y 2 + 6 xy 4
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Absolute Extrema (Maxs/Mins) Theorem. Extreme Value If f is a continuous function on a closed and bounded region R , then f has an absolute maximum AND an absolute minimum. Note In R 2 : 5
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How to find absolute extrema on an interval
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Unformatted text preview: [ a, b ] 1. Find critical points (C.P.) i.e. where f ( x ) = 0 or f ( x ) is undened. 2. Evaluate f at each C.P. and at the endpoints of the interval (i.e. the boundary). Largest value = Absolute Maximum Smallest value = Absolute Minimum In R 3 : How to nd Absolute Extrema 1. 2. 3. 4. 6 Example 1 Find the absolute extrema of f ( x, y ) = x 4 + y 3 +32 x-9 y over the rectangular plate-2 x 0,-2 y 0. 7 Example 2 Same function as Example 1, f ( x, y ) = x 4 + y 3 + 32 x-9 y over the region bounded by x = 2, y = 0, and y = 2 x . 8 Example 3 Find the extreme values of f ( x, y ) = xy on the unit disk, x 2 + y 2 1. 9...
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This note was uploaded on 03/03/2010 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.

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Extreme Values and Saddle Points - [ a, b ] 1. Find...

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