Practice 2 Solutions

Practice 2 Solutions - Math E-21a Fall 2009 Practice Exam...

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1 Math E-21a – Fall 2009 – Practice Exam #2 solutions 1) The level sets of a function (, ) f xy are shown. Determine the signs of the following derivatives at the point P. Indicate either +, –, or 0 in the spaces provided. x f = + y f = – xx f = – yy f = – xy f = + 2) Find all critical points of the function 22 2 2 4 2 2 1 4 fxy x y x y x  and classify each as a local maximum, a local minimum, or a saddle point. Solution : The stationary points (critical points) occur where the gradient f vanishes or, equivalently, where the partial derivatives 0 xy ff  . We compute: 2 2 2 0 48 4 ( 2 ) 0 x y fy x fx y y y x   Beginning with the second of these two equations, there are two cases: (a) where 0 y , or (b) where 2 x   . In case (a), substitution into the first equation yields 2 2 (1 1 ) 0 xx  , so 11 x   and we get the critical point 1 ,0 ) . In case (b), substitution into the first equation yields 2 24 2 2 21 8 2 (9 ) 0 yy y        , so either 3 y or 3 y and we get the two critical points (2 ,3 ) and ) . To test each of these, we calculate the Hessian matrix of second derivatives: 44 8 xx xy f yx yy y H yx    and evaluate this at each of the three critical points: 20 1 ) 03 6 f H  . det ( 11,0) 72 0 f H , so 1 ) gives either a local minimum or a local maximum, and xx f  means that there will be a local minimum at 1 ) . 2 ) 12 0 f H . det ( 2,3) 144 0 f H , so ) gives a saddle point. 2 ) 12 0 f H   . det ( 2, 3) 144 0 f H   , so ) gives a saddle point.
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2 3) Find the absolute maximum and minimum values of the function 22 (, ) 2 6 f xy x y x y  in the region whose boundary is the circle 160 xy  and identify the points where these maximum and minimum values occur. Solution : This problem requires two pieces of analysis. Inside the region (away from the boundary), this is an unconstrained optimization problem and is solved using critical point analysis. On the boundary curve, we use the method of Lagrange Multipliers.
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This note was uploaded on 02/10/2011 for the course MATH E-21a taught by Professor - during the Fall '09 term at Harvard.

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Practice 2 Solutions - Math E-21a Fall 2009 Practice Exam...

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