Unformatted text preview: Jignesh Patel due 11/21/2010 at 11:59pm EST.
. Assignment 8 MATH262, Fall 2010 1. (1 pt) Find the maximum value M of the function f (x, y) = x6 y9 (8 − x − y)9 on the region x ≥ 0, y ≥ 0, x + y ≤ 8. M= 2. (1 pt) Find the most economical dimensions of a closed rectangular box of volume 5 cubic units if the cost of the material per square unit for (i) the top and bottom is 2, (ii) the front and back is 5 and (iii) the other two sides is 7. Vertical edge length = Horizontal front and back edge length = Horizontal side edge length = 3. (1 pt) Find the maximum volume of a rectangular box that can be inscribed in the ellipsoid z2 x2 y2 + + = 1. 16 49 49 with sides parallel to the coordinate axes. Volume = 4. (1 pt) Find the maximum and minimum values of the function f (x, y) = 3x2 − 30xy + 3y2 − 7 on the disk x2 + y2 ≤ 9. Maximum = Minimum = 5. (1 pt) Find the maximum and minimum distance from the origin to a point on the curve 9x2 − 2xy + 9y2 = 5 in the (x, y) plane. Maximum distance = Minimum distance = 6. (1 pt) Consider the function f (x, y) = 5x − 11y + 4xy − 7x2 + 8y2 deﬁned in the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Find the maximum and minimum values of f and where they occur. The maximum value is and it occurs for x = and y = . and it occurs for x = and y = . The minimum value is 7. (1 pt) Consider the function f (x, y) = 5x3 − 6xy + 2y2 . Find the maximum and minimum values of f in the region deﬁned by the inequalities −1 ≤ x ≤ 1, 0 ≤ y ≤ 1 and where they occur. The maximum value is and it occurs for x= and y = . The minimum value is and it occurs for x= and y = . 8. (1 pt) Find the coordinates of the point (x, y, z) on the plane z = 4x + 4y + 3 which is closest to the origin. x= y= z= 9. (1 pt) Find the maximum and minimum values of f (x, y) = 6x + y on the ellipse x2 + 4y2 = 1 maximum value: minimum value: Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 1 ...
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