MATA33 Assignment 9 Winter 2022

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Department of Computer & Mathematical Sciences

MATA33

Assignment 9

Winter 2022

Problems: (Many and challenging 9)

1. Section 17.4, Pages 753 – 754 # 1 – 3, 6 – 8, 10 – 14, 16, 18 – 20, 22, 23, 24, 28, 29, 33, 34, 36.

2. A delivery company accepts only rectangular boxes who length plus ”girth” do not sum over 108 cm (The ”girth” of a rectangular box is de ned as the perimeter of a cross-section of the box). Find the dimensions of an acceptable box of largest volume.

3. In this question let f (x, y) = x2 _ y2 _ 2x + 4y + 6

(a) Use the critical point concepts and the second derivative test to nd out that f has a critical point at (1, 2) but no relative extrema there.

(b) Prove algebraically (i.e. not using calculus) that f has no relative extrema at (1, 2).

4. (a) Show that the critical point analysis and second derivative test provide no information about extrema of the function f (x, y) = x4 + y4

(b) Use algebra (and no calculus) to nd the local extrema of f in part (a). Prove also that the local extrema you nd in part (a) is actually absolute extrema.

5. (a) Repeat part (a) in Problem 4 for the function f (x, y) = x4 _ y4

(b) Use algebra (and no calculus) to show that f has no local extrema at the point (0, 0).

6. In this question let f (x, y) = x2 _ e(g2 1)

(a) Find all of the critical points of the function f

(b) Find all of the critical points of the function g(x) = f (x, x)

(c) What is surprising in your result for (b) compared to that of (a)

7. A rectangular box with no top is constructed from exactly 12m2 of material (i.e. there is no waste).

(a) With the length, width, and height represented by positive numbers x, y, and z respec- tively, show that the volume, V , of the box subject to the material constraint above is

xy(12 _ xy)

2x + 2y .

(b) Verify that if Vα (x, y) = Vg (x, y) = 0 then x = y .

(c) Re-read the paragraph entitled, ”Applications” on page 751 and convince yourself that there is a maximum volume of the box. Under this assumption, verify that the maxi- mum volume is 4m3 . (Note: the idea here is to not use the second-derivative test. That test would be quite complicated because of the second derivatives)

(d) Use (b) to write V as a function of x only and then use optimization methods from MATA32 to prove that the maximum value of the volume is 4m3 .

8. For each of the following functions of three variables, nd the critical point(s). Then for each critical point, use the second derivative test to determine whether it yields a local (i.e. relative) maximum, minimum, or saddle point, or that the behavior of the function is inconclusive at the critical point.

(a) f (x, y, z) = x3 + xy2 + x2 + y2 + 3z2

(b) f (x, y, z) = x3 + xz2 _ 3×2 + y2 + 2z2

(c) f (x, y, z) = x2y + y2 z + z2 _ 2x

(d) f (x, y, z) = xy _ xz