Extrema of Functions of Several Variables Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point for the function

(2) Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point for the function

(3) Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point for the function

(4) Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point for the function

(5) Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point for the function

(6) Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point for the function

(7) Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point for the function

(8) Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point for the function

(9) Find the absolute maximum and minimum values of the function on the disk

(10) Find the absolute maximum and minimum values of the function on the disk

(11) Find the absolute maximum and minimum values of the function on the triangular region with vertices and

(12) Find the absolute maximum and minimum values of the function on the square

(13) Find the absolute maximum and minimum values of the function on the square region with vertices and

(14) Find three positive numbers whose sum is 54 and whose product is as large as possible.

(15) A wire of length is cut into three pieces that are bent to form a circle, a square, and a equilateral triangle. How should the cuts be made to minimize the sum of the total area.?

(16) A rectangular box with no top is to have a fixed volume. What should its dimensions be if we want to use the least amount of material in its construction?

(17)  Let be the triangular region in the -plane with vertices and A plate in the shape of is heated so that the temperature at is



(in degrees Celsius). At what point in or on its boundary is maximized? What are the temperatures?

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