second_derivative_test_example - Note Second derivative...

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Note – Second derivative test [email protected] If U R n is a closed and bounded set and f : R n R is continuous on U , then f has a global maximum and minimum on U . To find these using the second derivative test, proceed as follows when U has a smooth boundary: Determine the critical points of f by solving f = 0. Discard critical points which are not contained in U . Determine f ( a ) for each critical point a U . Use the second derivative test to determine which values f ( a ) are local minima, maxima on U , and where the test fails. On the boundary ∂U of U , use the preceding steps to determine all local extremes. Compare all the values found to find the global minima and maxima. The second derivative test (the third step) can fail even though f ( a ) is a local maxi- mum or minimum. For example, if f ( x,y,z ) = x 4 + y 4 + z 4 , it is clear that f (0 , 0 , 0) = 0 is a local minimum of f , whereas the Hessian matrix is the all-zero matrix, and the second derivative test fails. The boundary ∂U
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This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.

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second_derivative_test_example - Note Second derivative...

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