A company produces x units of commodity A and y units of commodity B each hour.

Question

A company produces x units of commodity A and y units of commodity B each hour. The company can sell all of its units when commodity A sells for p=90-5x dollars per unit and commodity B sells for q=80-10y dollars per unit. The cost (in dollars) of producing these units is given by the joint-cost function C(x,y)=4xy+5. How much of commodity A and commodity B should be sold in order to maximize profit?

Commodity A __________ units

Commodity B__________ units

Consider the function: f(x,y)=ysqrtx-y^2-5x+19y

Find and classify all critical points of the function. If there are more blanks than critical points, leave the remaining entries blank.

fx=

fy=

fxx=

fxy=

fyy=

The critical point with the smallest x-coordinate is:

Explanation / Answer

The problem is somewhat ill defined. However, here is a solution:
P = profit = sale - production cost. Say, x is the number of commodity A units
produced (and sold) and y the same for commodity B. Then:
P(x,y) = x*(90-5x ) + y*(80-10y ) - (4xy+5).
For P maximum partial derivatives dP/dx=0 and dP/dy=0. Hence:
dP/dx = 90 - 10*x - 4*y = 0 => 5*x + 2y = 45
dP/dy = 80 - 20*y - 4*x = 0 => x + 5*y = 20.
Solution is: x = 185/23 = 8.04; y = 55/23 = 2.391.
Hence the optimal profit is for:
Commodity A: 8 units
Commodity B: 2 units sold.
Second derivative of P < 0 for (8.04, 2.391) proves that (8.04, 2.391) is maximum of P.

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