Multivariable Calculus

Multivariable Calculus

Lec 1 | MIT 18.02 Multivariable Calculus, Fall 2007

multivariable calculus?

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?

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

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