Chapter 14.4, Problem 26E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Profit Suppose that a firm produces two products, A and B, that sell for $a and$b, respectively, with the total cost of producing x units of A and y units of B equal to C(x, y). Show that when the profit from these products is maximized, ∂ C ∂ x ( x , y ) = a  and  ∂ C ∂ y ( x , y ) = b

To determine

To prove: The profit from the products, A and B, is maximized, when Cx(x,y)=a and Cy(x,y)=b where the total cost of producing to produce product A and B is C(x,y).

Explanation

Given Information:

The selling price of product A is $a and for product B is$b while the total cost of producing to produce product A and B is C(x,y).

Formula used:

The partial differentiation of z=f(x,y) with respect to x by holding y constant as:

âˆ‚âˆ‚x[f(x,y)]=fx(x,y)

The partial differentiate f(x,y) with respect to y by holding x constant as:

âˆ‚âˆ‚y[f(x,y)]=fy(x,y)

Proof:

The profit function is given by P(x,y)=R(x,y)âˆ’C(x,y), where R is the revenue function and C is the cost function.

If x units are produced of product A, each costs $a, and if y units are produced of product B, each costs$b, then the revenue is xa+yb.

Thus, the revenue function is R(x,y)=xa+yb.

Substitute xa+yb for R(x,y) in P(x,y)=R(x,y)âˆ’C(x,y).

P(x,y)=xa+ybâˆ’C(x,y)

Thus, the profit function is P(x,y)=xa+ybâˆ’C(x,y).

Differentiate P(x,y)=xa+ybâˆ’C(x,y) with respect to x by holding y constant,

âˆ‚Pâˆ‚x=âˆ‚âˆ‚x(xa+ybâˆ’C(x,y))=aâˆ’âˆ‚Câˆ‚x(x,y)

Equate the first derivative of function P(x,y)=xa+ybâˆ’C(x,y) with respect to x to zero

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