MATH 2177: Mathematical Topics for Engineers Homework #1 Due: Tuesday, September 9th (in recitation) Show an appropriate amount of work for each problem. Write your answers neatly and organized–keep your TA happy! Section 1.8 – Max/Min Problems • # 22: Find the critical points of the following function: f ( x, y) = x4 + y4 − 4x − 32y + 10. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. • # 33: Repeat the directions from #22 on the function: f ( x, y) = ye x − ey . • # 48: Find the absolute maximum and minimum value of the following function on given set R: f ( x, y) = x2 + y2 − 2x − 2y; R is the closed set bounded by the triangle with vertices (0, 0), (2, 0), and (0, 2). • # 58: Rectanglular boxes with a volume of 10 m3 are made of two materials. The material for the top and bottom of the box costs $10/m2 and the material for the sides of the box costs $1/m2 . What are the dimensions of the box that minimize the cost of the box? (over) MATH 2177 • Homework #1 Section 1.9 – Lagrange Multipliers • # 10: Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint: f ( x, y) = x − y subject to x2 + y2 − 3xy = 20. • # 15: Repeat the directions from #10 for: f ( x, y, z) = x + 3y − z subject to x2 + y2 + z2 = 4. • # 22: Repeat the directions from #10 for: f ( x, y, z) = x2 + y2 − z subject to z = 2x2 y2 + 1. • # 26: Find the rectangular box with a volume of 16 ft3 that has minimum surface area. • # 31: Find the point on the surface 4x + y − 1 = 0 closest to the point (1, 2, −3). (No additional response required, but think about how you know this is a minimum and not a maximum.)

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