12_8 - 12.8 Maxima and Minima Let f be a function with...

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12.8 Maxima and Minima Let f be a function with domain S , and let p 0 be a point in S . 1. f ( p 0 ) is a global maximum value of f on S if f ( p 0 ) f ( p ) for all p S . 2. f ( p 0 ) is a global minimum value of f on S if f ( p 0 ) f ( p ) for all p S . 3. f ( p 0 ) is a global extreme value of f on S if f ( p 0 ) is either a global maximum value or min global minimum value. If the above conditions only holds for p in N S , where N is a neighborhood of p 0 , then f ( p 0 ) is called a local maximum, local minimum or local extreme value of f . Max-Min Existence Theorem If f is continuous on a closed and bounded set S , then f attains both a (global) maximum value and a (global) minimum value. 1
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Where do extreme values occur? The extreme values can only occur at a critical point p 0 , where 1. p 0 is a boundary point of S . 2. p 0 is a stationary point, i.e. p 0 is an interior of S and f ( p 0 ) = 0 (where the tangent plane is horizontal). 3. p 0 is a singular point, i.e. p 0 is an interior of S and f is not differentiable at p 0 ( where the tangent plane has a sharp corner). 2
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Saddle point A point p 0 where f ( p 0 ) = 0 but f ( p 0 ) is not a local extreme value is called a saddle point . Example Let f ( x,y ) = 2 x 2 - y 2 . f x = 4 x = 0 x = 0 f y = - 2 y = 0 y = 0 but f (0 , 0) = 0 is not a local extreme value because f ( x, 0) = 2 x 2 > 0 and f (0 ,y ) = - y 2 < 0 for all x,y 6 = 0. 3
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Suppose that f ( x,y ) has continuous second par- tial derivatives in a neighborhood of ( x 0 ,y 0 ) and that f ( x 0 ,y 0 ) = 0 . Let D = D ( x 0 ,y 0 ) = f xx ( x 0 ,y 0 ) f yy ( x 0 ,y 0 ) - f xy ( x 0 ,y 0 ) 2 Then 1. if D > 0 and f xx ( x 0 ,y 0 ) < 0, then f ( x 0 ,y 0 ) is a local maximum value; 2. if D > 0 and f xx ( x 0 ,y 0 ) > 0, then f ( x 0 ,y 0 ) is a local minimum value; 3. if D < 0, then ( x 0 ,y 0 ) is a saddle point;
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This note was uploaded on 11/07/2010 for the course MATH 265 taught by Professor Gregorac during the Spring '08 term at Iowa State.

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12_8 - 12.8 Maxima and Minima Let f be a function with...

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