Exam 2 Solutions - Math E-21a Fall 2009 Exam#2 solutions 1...

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1 Math E-21a – Fall 2009 – Exam #2 solutions 1) Find all critical points of the function 3 2 ( , ) 6 6 3 2 f x y x y xy x y and, for each point, determine whether it is a local maximum, a local minimum, or a saddle point. Solution : The critical points (stationary points) occur where both partial derivatives vanish. We calculate: 2 2 2 2 3 6 6 3( 2 2) 0 (6 3) 2 6 5 ( 1)( 5) 0 2 6 3 0 2 6 3 x y f x y x y x x x x x x f y x y x These equations gives that either 1 x or 5 x . If we substitute these individually into the 2nd equation, we get the two critical points 3 2 (1, ) and 27 2 (5, ) . We then test each of these using the 2 nd derivative test. We first calculate the Hessian matrix of 2 nd derivatives: 6 6 6 2 xx xy f yx yy f f x H f f . For each of the critical points, we calculate: 3 2 6 6 (1, ) 6 2 f H , 3 2 det (1, ) 12 36 24 0 f H     ,so 3 2 (1, ) must be a saddle point. 27 2 30 6 (5, ) 6 2 f H , 27 2 det (5, ) 60 36 24 0 f H     and 30 0 xx f , so 27 2 (5, ) must be a local minimum. 2) Suppose that a student member of a Harvard committee has determined that the well-being of a work-study student is described by the function 0.6 0.4 ( , ) W x y x y , where x is her hourly salary and y is the number of hours per week she spends in class. The Administration has decreed that this committee may only recommend values of x and y that satisfy the constraint 2 40 x y . Use the Method of Lagrange Multipliers to find the values of
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