Exam 2 Solutions

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

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1 Math E-21a – Fall 2009 – Exam #2 solutions 1) Find all critical points of the function 32 (, ) 6 6 3 2 fxy x y x y 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: 22 36 6 3 (2 2 ) 0 (6 3) 2 6 5 ( 1)( 5) 0 263 0 x y fxy xy xx x x x x fy 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: 66 62 xx xy f yx yy ff x H     . For each of the critical points, we calculate: 3 2 (1, ) f H , 3 2 det 12 36 24 0 f H   ,so 3 2 (1, ) must be a saddle point. 27 2 30 6 (5, ) 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 Wxy 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 24 0  . Use the Method of Lagrange Multipliers to find the values of x and y that maximize student well-being according to this model.
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Exam 2 Solutions - Math E-21a Fall 2009 Exam #2 solutions...

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