# Problem 10a spring 2008 let f x y 2 3 x 3 1 3 y 3 xy

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Problem 10(a) - Spring 2008 Let f ( x , y ) = 2 3 x 3 + 1 3 y 3 - xy . Find all critical points of f ( x , y ).
Problem 10(b) - Spring 2008 Let f ( x , y ) = 2 3 x 3 + 1 3 y 3 - xy . Classify each critical point as a relative maximum , relative (local) minimum or saddle ; you do not need to calculate the function at these points, but your answer must be justified.
Problem 11 - Spring 2008 Use the method of Lagrange multipliers to determine all points ( x , y ) where the function f ( x , y ) = 2 x 2 + 4 y 2 + 16 has an extreme value (either a maximum or a minimum ) subject to the constraint 1 4 x 2 + y 2 = 4.
Problem 12 - Fall 2007 Find the x and y coordinates of all critical points of the function f ( x , y ) = 2 x 3 - 6 x 2 + xy 2 + y 2 and use the Second Derivative Test to classify them as local minima , local maxima or saddle points.
Problem 13(a) - Fall 2007 A hiker is walking on a mountain path. The surface of the mountain is modeled by z = 1 - 4 x 2 - 3 y 2 . The positive x -axis points to East direction and the positive y -axis points North . Suppose the hiker is now at the point P ( 1 4 , - 1 2 , 0) heading North, is she ascending or descending ? Justify your answers.