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second_derivative_test_example

# second_derivative_test_example - Note Second derivative...

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Note – Second derivative test 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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