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265Lesson12Problems

265Lesson12Problems - x 2 y 2 = 1 can be parametrized as x...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 12 PROBLEMS Problems are worth 20 points each. (1) There is one critical point for z = x 5 y + xy 5 + xy . Find it and classify it as a local maximum, local minimum, or a saddle point. (2) Find the critical point(s) of f ( x,y ) = 1 + ( x - y ) 2 + ( x - 1) 2 and classify each as a local maximum, local minimum, or a saddle point. (3) Find the critical point(s) of f ( x,y ) = xy 2 + x 2 y + 8 xy and classify each as a local maximum, local minimum, or a saddle point. (4) Determine the maximum value and the minimum value of f ( x,y ) = x 2 + y 2 - x - y on the closed unit disk D : x 2 + y 2 1. (Hint: Recall that the unit circle
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Unformatted text preview: x 2 + y 2 = 1 can be parametrized as x = cos t , y = sin t .) (5) Find the critical point(s) of f ( x,y ) = sin x + sin y + cos ( x + y ) for 0 ≤ x ≤ π 4 and ≤ y ≤ π 4 . Classify each as a local maximum, local minimum, or a saddle point. (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) A rectangular tank with a bottom and sides but no top is to have volume 500 cubic feet. Determine the dimensions (length, width, height) with the smallest possible surface area. 1...
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