# 4.7 Maxima/minima problems  (Page 6/10)

 Page 6 / 10

Use the problem-solving strategy for finding absolute extrema of a function to find the absolute extrema of the function

$f\left(x,y\right)=4{x}^{2}-2xy+6{y}^{2}-8x+2y+3$

on the domain defined by $0\le x\le 2$ and $-1\le y\le 3.$

The absolute minimum occurs at $\left(1,0\right)\text{:}$ $f\left(1,0\right)=-1.$

The absolute maximum occurs at $\left(0,3\right)\text{:}$ $f\left(0,3\right)=63.$

## Chapter opener: profitable golf balls

Pro- $\text{T}$ company has developed a profit model that depends on the number x of golf balls sold per month (measured in thousands), and the number of hours per month of advertising y , according to the function

$z=f\left(x,y\right)=48x+96y-{x}^{2}-2xy-9{y}^{2},$

where $z$ is measured in thousands of dollars. The maximum number of golf balls that can be produced and sold is $50,000,$ and the maximum number of hours of advertising that can be purchased is $25.$ Find the values of $x$ and $y$ that maximize profit, and find the maximum profit.

Using the problem-solving strategy, step $1$ involves finding the critical points of $f$ on its domain. Therefore, we first calculate ${f}_{x}\left(x,y\right)$ and ${f}_{y}\left(x,y\right),$ then set them each equal to zero:

$\begin{array}{ccc}\hfill {f}_{x}\left(x,y\right)& =\hfill & 48-2x-2y\hfill \\ \hfill {f}_{y}\left(x,y\right)& =\hfill & 96-2x-18y.\hfill \end{array}$

Setting them equal to zero yields the system of equations

$\begin{array}{ccc}\hfill 48-2x-2y& =\hfill & 0\hfill \\ \hfill 96-2x-18y& =\hfill & 0.\hfill \end{array}$

The solution to this system is $x=21$ and $y=3.$ Therefore $\left(21,3\right)$ is a critical point of $f.$ Calculating $f\left(21,3\right)$ gives $f\left(21,3\right)=48\left(21\right)+96\left(3\right)-{21}^{2}-2\left(21\right)\left(3\right)-9{\left(3\right)}^{2}=648.$

The domain of this function is $0\le x\le 50$ and $0\le y\le 25$ as shown in the following graph.

${L}_{1}$ is the line segment connecting $\left(0,0\right)$ and $\left(50,0\right),$ and it can be parameterized by the equations $x\left(t\right)=t,y\left(t\right)=0$ for $0\le t\le 50.$ We then define $g\left(t\right)=f\left(x\left(t\right),y\left(t\right)\right)\text{:}$

$\begin{array}{cc}\hfill g\left(t\right)& =f\left(x\left(t\right),y\left(t\right)\right)\hfill \\ & =f\left(t,0\right)\hfill \\ & =48t+96\left(0\right)-{y}^{2}-2\left(t\right)\left(0\right)-9{\left(0\right)}^{2}\hfill \\ & =48t-{t}^{2}.\hfill \end{array}$

Setting ${g}^{\prime }\left(t\right)=0$ yields the critical point $t=24,$ which corresponds to the point $\left(24,0\right)$ in the domain of $f.$ Calculating $f\left(24,0\right)$ gives $576.$

${L}_{2}$ is the line segment connecting  and $\left(50,25\right),$ and it can be parameterized by the equations $x\left(t\right)=50,y\left(t\right)=t$ for $0\le t\le 25.$ Once again, we define $g\left(t\right)=f\left(x\left(t\right),y\left(t\right)\right)\text{:}$

$\begin{array}{cc}\hfill g\left(t\right)& =f\left(x\left(t\right),y\left(t\right)\right)\hfill \\ & =f\left(50,t\right)\hfill \\ & =48\left(50\right)+96t-{50}^{2}-2\left(50\right)t-9{t}^{2}\hfill \\ & =-9{t}^{2}-4t-100.\hfill \end{array}$

This function has a critical point at $t=-\frac{2}{9},$ which corresponds to the point $\left(50,-\frac{2}{9}\right).$ This point is not in the domain of $f.$

${L}_{3}$ is the line segment connecting $\left(0,25\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(50,25\right),$ and it can be parameterized by the equations $x\left(t\right)=t,y\left(t\right)=25$ for $0\le t\le 50.$ We define $g\left(t\right)=f\left(x\left(t\right),y\left(t\right)\right)\text{:}$

$\begin{array}{cc}\hfill g\left(t\right)& =f\left(x\left(t\right),y\left(t\right)\right)\hfill \\ & =f\left(t,25\right)\hfill \\ & =48t+96\left(25\right)-{t}^{2}-2t\left(25\right)-9\left({25}^{2}\right)\hfill \\ & =\text{−}{t}^{2}-2t-3225.\hfill \end{array}$

This function has a critical point at $t=-1,$ which corresponds to the point $\left(-1,25\right),$ which is not in the domain.

${L}_{4}$ is the line segment connecting $\left(0,0\right)\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}\left(0,25\right),$ and it can be parameterized by the equations $x\left(t\right)=0,y\left(t\right)=t$ for $0\le t\le 25.$ We define $g\left(t\right)=f\left(x\left(t\right),y\left(t\right)\right)\text{:}$

$\begin{array}{cc}\hfill g\left(t\right)& =f\left(x\left(t\right),y\left(t\right)\right)\hfill \\ & =f\left(0,t\right)\hfill \\ & =48\left(0\right)+96t-{\left(0\right)}^{2}-2\left(0\right)t-9{t}^{2}\hfill \\ & =96t-{t}^{2}.\hfill \end{array}$

This function has a critical point at $t=\frac{16}{3},$ which corresponds to the point $\left(0,\frac{16}{3}\right),$ which is on the boundary of the domain. Calculating $f\left(0,\frac{16}{3}\right)$ gives $256.$

We also need to find the values of $f\left(x,y\right)$ at the corners of its domain. These corners are located at $\left(0,0\right),\left(50,0\right),\left(50,25\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(0,25\right)\text{:}$

$\begin{array}{ccc}\hfill f\left(0,0\right)& =\hfill & 48\left(0\right)+96\left(0\right)-{\left(0\right)}^{2}-2\left(0\right)\left(0\right)-9{\left(0\right)}^{2}=0\hfill \\ \hfill f\left(50,0\right)& =\hfill & 48\left(50\right)+96\left(0\right)-{\left(50\right)}^{2}-2\left(50\right)\left(0\right)-9{\left(0\right)}^{2}=-100\hfill \\ \hfill f\left(50,25\right)& =\hfill & 48\left(50\right)+96\left(25\right)-{\left(50\right)}^{2}-2\left(50\right)\left(25\right)-9{\left(25\right)}^{2}=-5825\hfill \\ \hfill f\left(0,25\right)& =\hfill & 48\left(0\right)+96\left(25\right)-{\left(0\right)}^{2}-2\left(0\right)\left(25\right)-9{\left(25\right)}^{2}=-3225.\hfill \end{array}$

The maximum critical value is $648,$ which occurs at $\left(21,3\right).$ Therefore, a maximum profit of $\text{}648,000$ is realized when $21,000$ golf balls are sold and $3$ hours of advertising are purchased per month as shown in the following figure.

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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preparation of nanomaterial
how did you get the value of 2000N.What calculations are needed to arrive at it
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