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test1-a

# X y 0 and the discriminant d f xx x y f yy x y f xy x

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x 0 , y 0 ) = 0 and the discriminant d = f xx ( x 0 , y 0 ) f yy ( x 0 , y 0 ) - f xy ( x 0 , y 0 ) 2 is nonzero (i.e., the Second Derivative Test does not fail). a) Prove that if u ( x, y ) satisFes the Laplace equation (no boundary conditions are assumed), then u ( x, y ) has no nondegenerate minima or maxima, i.e., all nondegenerate critical points are saddle points. b) Prove that if u ( x, t ) satisFes the heat equation (no boundary conditions are assumed), then u ( x, t ) has no nondegenerate minima or maxima, i.e., all nondegenerate critical points are saddle points.

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c) Provide an example of a function u ( x, t ) that satisFes the wave equation AND it has a relative minimum and/or a relative maximum. As with the previous problems, no boundary conditions are assumed. 7) (10 points) You showed in Homework 2 that the ±ourier series for f ( x ) = | x | on - π x < π is: π 2 - 4 π s k =0 1 (2 k + 1) 2 cos(2 k + 1) x a) Use Parseval’s Identity on f ( x ) = | x | to come up with a neat inFnite sum identity. b) ±ollowing the process you performed in 2.5-7 b) and c), use the value of ζ (4) to come up with the same inFnite sum identity.
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x y 0 and the discriminant d f xx x y f yy x y f xy x y 2...

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