Midterm2 Solutions.pdf - 2(a Take the function f(x y = x3 y 3 points of f(x y rf =(3x2(0.1 9xy 1 Find and classify critical 9y 3y 2 9x Critical points

Midterm2 Solutions.pdf - 2(a Take the function f(x y = x3 y...

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2(a). Take the function f ( x, y ) = x 3 + y 3 - 9 xy + 1. Find and classify critical points of f ( x, y ). (0.1) r f = (3 x 2 - 9 y, 3 y 2 - 9 x ) Critical points must satisfiy r f = 0, namely (0.2) 3 x 2 = 9 y, 3 y 2 = 9 x Solve it we get 2 solutions: ( x, y ) = (0 , 0) or ( x, y ) = (3 , 3) To classify the critical points, we need to analyze the Hessian matrix of f. (0.3) f xx = 6 x, f yy = 6 y, f xy = - 9 So the Hessian matrix is (0.4) 6 x - 9 - 9 6 y At the point (0 , 0), the Hessian is 0 - 9 - 9 0 , which is indefinite, it’s a saddle point. At the point (3 , 3), the Hessian is 18 - 9 - 9 18 , which is indefinite, it’s a local minimum point. 1
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MATH 202 Midterm 2, Problem 2(b) We want to find the maximum and minimum values of f ( x, y ) in the triangle R with the vertices (4 , 0), (0 , 4), and (4 , 4). To achieve this, we need to check all critical points on the interior, and all critical points on the boundary. The critical points on the interior were given to us in the previous problem. The only one is at (3 , 3). The boundary can be decomposed into three curves l 1 , l 2 , and l 3 , plus three corners (the points (4 , 0), (0 ,
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