multivariable calculus?

Question

it's a profit question that I have to solve:

Suppose that the profit (in hundreds of dollars) of a certain firm is approximated by

P(x,y) = 1000 + 24x - x^2 + 80y - y^2

where x is the cost of a unit of labor and y is the cost of a unit of goods. Find values of x and y that maximize profit. Find the maximum profit.

can anyone please help me out?

Answer 1

You have to find the absolute maxima for both variables.
So, find dP(x) and dP(y)

Then find the critical points, fx(a,b)=0 and fy(a,b)=0

and solve d=fxx(a,b)*fyy(a,b) - [fxy(a,b)]^2

if d>0 and fxx(a,b)>0 then f has a relative min at (a,b)
if d>0 and fxx(a,b)<0 then f has a relative max at (a,b)
if d<0 then (a,b,f(a,b)) is a saddle point
if d=0 then inconclusive

Answer 2

Hello,

y = 1000 + 24x - x^2 + 80y - y^2

Take the differentials of x and y giving us:

dy = 24dx -2xdx + 80dy -2ydy

Put all the dy's on the left and we have

dy-80dy+2ydy = 24dx - 2xdx

thats

-79dy + 2ydy = 24dx -2xdx

factor out dy on the left and a dx on the right

(-79 + 2y) dy = (24 - 2x)dx

solve for dy
dy = (24-2x)dx/(2y-79)

divide by dx

dy/dx = (24-2x)/(2y-79)

to find the max or min set this = 0

0 = (24-2x)/(2y-79)

mul both sides by (2y-79) and we have 0 = 24 -2x so x = 12.


Put x = 12 back into the original giving us.

y = 1000 + 24*12 -12^2 + 80y -y^2

Put every thing on the left side

y^2 -79y - 1144 = 0

Put this into the quadratic equation and find y is approx 91.50.

Now put x = 12 and y = 91.5 into the original and get about 91.75.

Hope this helps!

Answer 3

i became into going to declare that that's feasible you ought to bypass in case you have been enormously drone like as a student and took each and every thing anybody pronounced without question and as fact, yet extremely that would not even assist you to. all people may be previous puzzled in this direction in the event that they don't comprehend what a spinoff is, not to show an critical. lots of the direction is vectors, yet something is a project of in case you are able to truly comprehend the calculus sufficient to jot down down and choose which critical is the correct one needed for particular applications. are you able to try this? not likely in case you do not even comprehend integrals. Even pupils that have taken calc a million and a pair of somewhat carry close the belief, calc 3 would nicely be an exceedingly difficult class in case your college is hard sufficient. It merely could not artwork out even with the incontrovertible fact that for many all people. decide your schedule and make some sacrifices, failing a class may be stupid this early on.

Answer 4

1.Take partial derivatives of P wrt x and y.

2.Set them to zero.

3. Solve the resulting system of 2 equations with 2 unknowns
this is where P(x,y) gets its maximum

4. Plug in these values for x and y into P to get the maximum.

Answer 5

P(x,y) = 1000 + 24x - x^2 + 80y - y^2
When ∂P/∂x = 24 - 2x = 0 AND ∂P/∂y = 80 - 2y = 0
you will have a minimum or a maximum
x,y = (12,40)
P(12,40) = 1,000 + 24*12 - 144 + 80*40 - 1600
P(12,40) = $2,744


x\y . . 39 . . 40 . . 41
11 2,742 2,743 2,742
12 2,743 2,744 2,743
13 2,742 2,743 2,742

Answer 6

sum of x and y >or =77