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Quiz 9: Maxima and minima of functions of two variables

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Question 1

Find the critical point and its nature for the function f(x,y) = x2 - 2x + 2y2 + 4y- 2 .

a)
(1,1), a maximum
   b)
(1,-1), a maximum
c)
(1,1), a minimum
   d)
(1,-1), a minimum

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
∂f- ∂f- ∂x = 2x - 2 ⇒ ∂x = 0 when x = 1.
∂f-= 4y + 4 ⇒ ∂f-= 0 ∂y ∂y when y = -1.
f = -5 at (1,-1).
The surface is concave up at all points.

Question 2

The function f(x,y) = 1+ 2x+ 8y - x2 - 2y2 has one critical point. Determine its position and nature.

a)
(1,2), a maximum
   b)
(1,-2), a maximum
c)
(1,2), a minimum
   d)
(1,-2), a minimum

 

Your answer is correct.
∂f-= 2- 2x ⇒ ∂f-= 0 ∂x ∂x when x = 1.
∂f-= 8- 4y ⇒ ∂f-= 0 ∂y ∂y when y = 2.
f = 10 at (1,2).
The surface is concave down at all points.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 3

How many critical points has the function 3 1 1 f(x,y) =-y2 + --y3 --- y4 - x2 4 24 32 ?

a)
0
   b)
1
c)
2
   d)
3

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
∂f- = 2x, ∂x ∂f- = 3y+ 1y2 - 1y3, ∂y 2 8 8 1 = -8 y(y + 3)(y- 4).
therefore the critical points are at (0,-3),(0,0) and (0,4).

Question 4

Find the horizontal tangent plane to the surface 2 1 3 2 z = 3x - 2x + 3x - y when x = 1.

a)
y = 0
   b)
23 z = --- 3
c)
4 z = 3
  d)
z = -18

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
∂z ∂y- = - 2y, ∂z --- = 3 - 4x+ x2, ∂x = (3 - x)(1- x).
Therefore, the critical points are at (1,0) and (3,0). Thus the horizontal tangent plane at (1,0) is 1 4 z = 3- 2+ - - 0 = - 3 3 .
Not correct. Choice (d) is false.

Question 5

Which equation most closely corresponds to the sketch of the surface below ?

PIC
a)
z = 5 - (x + 2)2 - 2(y + 3)2
  b)
z = 5- (x- 2)2 - 2(y- 3)2
c)
2 2 z = 5+ (x+ 2) + 2(y+ 3)
  d)
z = 5+ (x- 2)2 + 2(y- 3)2

 

Not correct. Choice (a) is false.
Your answer is correct.
There is a critical point at (2,3) with z = 5. The function needs to be concave down. Hence the correct answer is B.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 6

What does 5 ∑ 2k+ 1 k=1 mean ?

a)
2(1 + 2 + 3 + 4 + 5) + 1
   b)
3 + 11
c)
3 + 5 + 7 + 9 + 11
   d)
None of the above

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
∑5 2k +1 = (2× 1+ 1)+ (2× 2 + 1) + (2 × 3+ 1) k=1 + (2× 4 +1) +(2 × 5+ 1) = 3+ 5 + 7+ 9+ 11.
(1)
Not correct. Choice (d) is false.

Question 7

Which of the following represents 2 + 7 + 14 + 23 + 34 in summation notation ?

a)
∑5 2 k + 1 k=1
  b)
∑5 (k+ 1)2 - 2 k=1
c)
∑5 k + 1 k=1
  d)
∑5 2 (k- 2) + 1 k=1

 

Not correct. Choice (a) is false.
Your answer is correct.
∑5 2 (k+ 1) - 2 = (4- 2)+ (9- 2)+ (16- 2) k=1 + (25- 2)+ (36- 2) = 2 +7 + 14+ 23+ 34.
(2)
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 8

An airline will only accept luggage whose “linear length” height + width + length is at most 200cm. Assume that you will choose a suitcase with a standard box shape. Which of the following correctly represents the problem of finding what is the maximum volume that your suitcase can be?

a)
Find h, w, d that maximise V (h,w,d) = hwd for 0 ≤ h ≤ 200cm, 0 ≤ w ≤ 200cm, 0 ≤ d ≤ 200cm.
   b)
Find the maximum possible value of V (h,w,d) = hwd for 0 ≤ h ≤ 200cm, 0 ≤ w ≤ 200cm, 0 ≤ d ≤ 200cm.
c)
Find the maximum of V (h,w) = hw(200 - w - h) where h,w ≥ 0.
   d)
Find the maximum of V (h,w) = 200hw where h,w ≥ 0

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.

Question 9

An airline will only accept luggage whose “linear length” height + width + length is at most 200cm. Assume that you will choose a suitcase with a standard box shape. What is the maximum volume that your suitcase can be? Hint: The solution to 200x - y2 - 2xy = 0 and 200y - x2 - 2xy = 0, x,y ≥ 0 is x = y = 200- 3

a)
(2003-)3cm3 = 2.96 × 105cm3
   b)
2030cm3 = 66.667cm3 to three decimal places.
c)
3 2003-cm3 = 2.667 × 106cm3
   d)
200cm3

 

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 10

An airline will only accept luggage whose “linear length” height + width + length is at most 200cm. Assume that these are measured at the widest part of the piece of luggage. Could you carry more using a spherical suitcase than using a box-shaped suitcase as described in questions 8 and 9?

a)
Yes
   b)
No

 

Not correct. Choice (a) is false.
Your answer is correct.
A cube with sides of length 200 3cm will have volume (200- 3)3cm3 = 296,000cm3 approximately. A sphere with the same diameter would have radius 2060cm, so its volume would be 4π3(2060)3 = 155,140cm3 approximately.
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