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Unformatted text preview: given by f at this point. 4. (14 points) Find lim x , y ( 29 0,0 ( 29 xy 3 x 2 + y 6 , or explain in detail why this limit does not exist. 5. (14 points) Find and classify all critical points of f x , y ( 29 = -x 4 + 4 xy 3-2 y 2 + 1. 6. (14 points) Use differentials to approximate the length of the hypotenuse of a right triangle whose other two sides are lengths 4.2 and 2.9. 7. (14 points) Suppose the sum of the lengths of all the edges of a rectangular box is given to be a constant, say p . Use Lagrange multipliers to prove (not just guess) that in order to have the largest possible volume, this box must actually be a cube. What is its maximum volume? 8. (14 points) Find the absolute extrema of f x , y ( 29 = 2 x 3 + y 4 over the square region whose boundary connects the points 1,1 ( 29 , 1,-1 ( 29 ,-1,-1 ( 29 ,and-1,1 ( 29 in the xy-plane....
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- Winter '08