## H1011 Quizzes

Quiz 9: Maxima and minima of functions of two variables
Question 1 Questions
Find the critical point and its nature for the function $f\left(x,y\right)={x}^{2}-2x+2{y}^{2}+4y-2$.
 a) (1,1), a maximum b) (1,-1), a maximum c) (1,1), a minimum d) (1,-1), a minimum

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$\frac{\partial f}{\partial x}=2x-2⇒\frac{\partial f}{\partial x}=0$ when $x=1$.
$\frac{\partial f}{\partial y}=4y+4⇒\frac{\partial f}{\partial y}=0$ when $y=-1$.
$f=-5$ at (1,-1).
The surface is concave up at all points.
The function $f\left(x,y\right)=1+2x+8y-{x}^{2}-2{y}^{2}$ 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

Choice (a) is correct!
$\frac{\partial f}{\partial x}=2-2x⇒\frac{\partial f}{\partial x}=0$ when $x=1$.
$\frac{\partial f}{\partial y}=8-4y⇒\frac{\partial f}{\partial y}=0$ when $y=2$.
$f=10$ at (1,2).
The surface is concave down at all points.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
How many critical points has the function $f\left(x,y\right)=\frac{3}{4}{y}^{2}+\frac{1}{24}{y}^{3}-\frac{1}{32}{y}^{4}-{x}^{2}$ ?
 a) 0 b) 1 c) 2 d) 3

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$\begin{array}{rcll}\frac{\partial f}{\partial x}& =& 2x,& \text{}\\ \frac{\partial f}{\partial y}& =& \frac{3}{2}y+\frac{1}{8}{y}^{2}-\frac{1}{8}{y}^{3},& \text{}\\ & =& -\frac{1}{8}y\left(y+3\right)\left(y-4\right).& \text{}\end{array}$ therefore the critical points are at (0,-3),(0,0) and (0,4).
Find the horizontal tangent plane to the surface $z=3x-2{x}^{2}+\frac{1}{3}{x}^{3}-{y}^{2}$ when $x=1$.
 a) $y=0$ b) $z=-\frac{23}{3}$ c) $z=\frac{4}{3}$ d) $z=-18$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$\begin{array}{rcll}\frac{\partial z}{\partial y}& =& -2y,& \text{}\\ \frac{\partial z}{\partial x}& =& 3-4x+{x}^{2},& \text{}\\ & =& \left(3-x\right)\left(1-x\right).& \text{}\end{array}$ Therefore, the critical points are at (1,0) and (3,0). Thus the horizontal tangent plane at (1,0) is $z=3-2+\frac{1}{3}-0=\frac{4}{3}$.
Choice (d) is incorrect
Which equation most closely corresponds to the sketch of the surface below ?
 a) $z=5-{\left(x+2\right)}^{2}-2{\left(y+3\right)}^{2}$ b) $z=5-{\left(x-2\right)}^{2}-2{\left(y-3\right)}^{2}$ c) $z=5+{\left(x+2\right)}^{2}+2{\left(y+3\right)}^{2}$ d) $z=5+{\left(x-2\right)}^{2}+2{\left(y-3\right)}^{2}$

Choice (a) is incorrect
Choice (b) 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.
Choice (c) is incorrect
Choice (d) is incorrect
What does ${\sum }_{k=1}^{5}2k+1$ mean ?
 a) $2\left(1+2+3+4+5\right)+1$ b) $3+11$ c) $3+5+7+9+11$ d) None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$\begin{array}{cc}& \sum _{k=1}^{5}2k+1=\left(2×1+1\right)+\left(2×2+1\right)+\left(2×3+1\right)\\ & +\left(2×4+1\right)+\left(2×5+1\right)=3+5+7+9+11.& \text{(1)}\end{array}$
Choice (d) is incorrect
Which of the following represents $2+7+14+23+34$ in summation notation ?
 a) ${\sum }_{k=1}^{5}{k}^{2}+1$ b) ${\sum }_{k=1}^{5}{\left(k+1\right)}^{2}-2$ c) ${\sum }_{k=1}^{5}k+1$ d) ${\sum }_{k=1}^{5}{\left(k-2\right)}^{2}+1$

Choice (a) is incorrect
Choice (b) is correct!
$\begin{array}{cc}& \sum _{k=1}^{5}{\left(k+1\right)}^{2}-2=\left(4-2\right)+\left(9-2\right)+\left(16-2\right)\\ & +\left(25-2\right)+\left(36-2\right)=2+7+14+23+34.& \text{(2)}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
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 $V\left(h,w,d\right)=hwd$ for $0\le h\le 200cm$, $0\le w\le 200cm$, $0\le d\le 200cm$. b) Find the maximum possible value of $V\left(h,w,d\right)=hwd$ for $0\le h\le 200cm$, $0\le w\le 200cm$, $0\le d\le 200cm$. c) Find the maximum of $V\left(h,w\right)=hw\left(200-w-h\right)$ where $h,w\ge 0$. d) Find the maximum of $V\left(h,w\right)=200hw$ where $h,w\ge 0$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
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 $200x-{y}^{2}-2xy=0$ and $200y-{x}^{2}-2xy=0$, $x,y\ge 0$ is $x=y=\frac{200}{3}$
 a) ${\left(\frac{200}{3}\right)}^{3}c{m}^{3}=2.96×1{0}^{5}c{m}^{3}$ b) $\frac{200}{3}c{m}^{3}=66.667c{m}^{3}$ to three decimal places. c) $\frac{20{0}^{3}}{3}c{m}^{3}=2.667×1{0}^{6}c{m}^{3}$ d) $200c{m}^{3}$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
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

Choice (a) is incorrect
Choice (b) is correct!
A cube with sides of length $\frac{200}{3}cm$ will have volume ${\left(\frac{200}{3}\right)}^{3}c{m}^{3}=296,000c{m}^{3}$ approximately. A sphere with the same diameter would have radius $\frac{200}{6}cm$, so its volume would be $\frac{4\pi }{3}{\left(\frac{200}{6}\right)}^{3}=155,140c{m}^{3}$ approximately.