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# tut7 - P from selling a product depends on advertising and...

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MATH 237 Tutorial 7 Wednesday, June 28th, 2006 Critical Points and Optimization Problems 1) Let f ( x, y ) = xy 2 + x 2 y - 4 xy . a) Find and classify all critical points of f . b) Find the extreme values of f on the closed triangle bounded by the lines x = 0 , y = 0 and x + y = 2. 2) Suppose that f ( x, y ) = x 2 + y 2 - x 2 y 2 . a) Find and classify all critical points of f . b) Find the maximum and minimum values of f on the square S = { ( x, y ) | 0 x 2 , 0 y 2 } . 3) Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2 y + 3 z = 6. 4) Find the minimum value of f ( x, y ) = x + 8 y + 1 xy in the first quadrant x > 0 , y > 0. How do you know a minimum exists? 5) Use Lagrange multipliers to find the maximum value of x + y + z on the ellipsoid x 2 + y 2 + 1 2 z 2 = 1. 6) The profit,
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Unformatted text preview: P , from selling a product depends on advertising and is given by P ( x, y ) = 100 x 1 / 5 y 4 / 5 where x is the number of times the ad for the product appears in a newspaper and y is the number of times the ad appears on TV. If \$ 600,000 is available for advertising, and the cost of each appearance of a newspaper ad is \$ 3,000 and of a TV ad is \$ 4,000, ±nd x and y that will maximize the pro±t. 7) A silo is in the shape of a cylinder topped with a cone. If the radius of each is 6 m and the total surface area is 200 m 2 (excluding the base), what are the heights of the cylinder and cone that maximize the volume enclosed by the silo?...
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