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

f ( x , y ) = 4 x 2 2 x y + 6 y 2 8 x + 2 y + 3

on the domain defined by 0 x 2 and −1 y 3 .

The absolute minimum occurs at ( 1 , 0 ) : f ( 1 , 0 ) = −1 .

The absolute maximum occurs at ( 0 , 3 ) : f ( 0 , 3 ) = 63 .

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Chapter opener: profitable golf balls

A basket full of golf balls.
(credit: modification of work by oatsy40, Flickr)

Pro- 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 ( x , y ) = 48 x + 96 y x 2 2 x y 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 ( x , y ) and f y ( x , y ) , then set them each equal to zero:

f x ( x , y ) = 48 2 x 2 y f y ( x , y ) = 96 2 x 18 y .

Setting them equal to zero yields the system of equations

48 2 x 2 y = 0 96 2 x 18 y = 0 .

The solution to this system is x = 21 and y = 3 . Therefore ( 21 , 3 ) is a critical point of f . Calculating f ( 21 , 3 ) gives f ( 21 , 3 ) = 48 ( 21 ) + 96 ( 3 ) 21 2 2 ( 21 ) ( 3 ) 9 ( 3 ) 2 = 648 .

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

A rectangle is drawn in the first quadrant with one corner at the origin, horizontal length 50, and height 25. This rectangle is marked D, and the sides are marked in counterclockwise order from the side overlapping the x axis L1, L2, L3, and L4.
Graph of the domain of the function f ( x , y ) = 48 x + 96 y x 2 2 x y 9 y 2 .

L 1 is the line segment connecting ( 0 , 0 ) and ( 50 , 0 ) , and it can be parameterized by the equations x ( t ) = t , y ( t ) = 0 for 0 t 50 . We then define g ( t ) = f ( x ( t ) , y ( t ) ) :

g ( t ) = f ( x ( t ) , y ( t ) ) = f ( t , 0 ) = 48 t + 96 ( 0 ) y 2 2 ( t ) ( 0 ) 9 ( 0 ) 2 = 48 t t 2 .

Setting g ( t ) = 0 yields the critical point t = 24 , which corresponds to the point ( 24 , 0 ) in the domain of f . Calculating f ( 24 , 0 ) gives 576 .

L 2 is the line segment connecting and ( 50 , 25 ) , and it can be parameterized by the equations x ( t ) = 50 , y ( t ) = t for 0 t 25 . Once again, we define g ( t ) = f ( x ( t ) , y ( t ) ) :

g ( t ) = f ( x ( t ) , y ( t ) ) = f ( 50 , t ) = 48 ( 50 ) + 96 t 50 2 2 ( 50 ) t 9 t 2 = −9 t 2 4 t 100 .

This function has a critical point at t = 2 9 , which corresponds to the point ( 50 , 2 9 ) . This point is not in the domain of f .

L 3 is the line segment connecting ( 0 , 25 ) and ( 50 , 25 ) , and it can be parameterized by the equations x ( t ) = t , y ( t ) = 25 for 0 t 50 . We define g ( t ) = f ( x ( t ) , y ( t ) ) :

g ( t ) = f ( x ( t ) , y ( t ) ) = f ( t , 25 ) = 48 t + 96 ( 25 ) t 2 2 t ( 25 ) 9 ( 25 2 ) = t 2 2 t 3225 .

This function has a critical point at t = −1 , which corresponds to the point ( −1 , 25 ) , which is not in the domain.

L 4 is the line segment connecting ( 0 , 0 ) to ( 0 , 25 ) , and it can be parameterized by the equations x ( t ) = 0 , y ( t ) = t for 0 t 25 . We define g ( t ) = f ( x ( t ) , y ( t ) ) :

g ( t ) = f ( x ( t ) , y ( t ) ) = f ( 0 , t ) = 48 ( 0 ) + 96 t ( 0 ) 2 2 ( 0 ) t 9 t 2 = 96 t t 2 .

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

We also need to find the values of f ( x , y ) at the corners of its domain. These corners are located at ( 0 , 0 ) , ( 50 , 0 ) , ( 50 , 25 ) and ( 0 , 25 ) :

f ( 0 , 0 ) = 48 ( 0 ) + 96 ( 0 ) ( 0 ) 2 2 ( 0 ) ( 0 ) 9 ( 0 ) 2 = 0 f ( 50 , 0 ) = 48 ( 50 ) + 96 ( 0 ) ( 50 ) 2 2 ( 50 ) ( 0 ) 9 ( 0 ) 2 = −100 f ( 50 , 25 ) = 48 ( 50 ) + 96 ( 25 ) ( 50 ) 2 2 ( 50 ) ( 25 ) 9 ( 25 ) 2 = −5825 f ( 0 , 25 ) = 48 ( 0 ) + 96 ( 25 ) ( 0 ) 2 2 ( 0 ) ( 25 ) 9 ( 25 ) 2 = −3225 .

The maximum critical value is 648 , which occurs at ( 21 , 3 ) . Therefore, a maximum profit of $ 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.

The function f(x, y) = 48x + 96y – x2 – 2xy – 9y2 is shown with maximum point at (21, 3, 648). The shape is a plane curving from near the origin down to (50, 25).
The profit function f ( x , y ) has a maximum at ( 21 , 3 , 648 ) .
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Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
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I think
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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analytical skills graphene is prepared to kill any type viruses .
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what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
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what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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why we need to study biomolecules, molecular biology in nanotechnology?
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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