Question

A company produces x units of product A and y units of product B (both in hundreds per month). The monthly profit equation (in thousands of dollars) is given by

P(x, y) = -4x2 + 4xy - 3y2 + 4x + 10y + 81

(A) Find Px(1, 3) and interpret the results.

(B) How many of each product should be produced each month to maximize profit? What is the maximum profit?

Answer 1

P(x, y) = -4x2 + 4xy – 3y2 + 4x + 10y + 81

 

A)

Px(x, y) = -8x + 4y + 4

Px(1, 3) = -8 + 12 + 4

              = -4 + 2

             = 8

 

B)

Px(x, y) = -8x + 4y + 4

Py(x, y) = 4x – 6y + 10

 

Px = 0 ⇒  -2x + y = -1

Py = 0 ⇒  2x – 3y = -5

-2y = -6

⇒ y = 3

 

2x – 6 = -5

⇒ 2x = 1

⇒ (1/2, 3)

 

Pxx = -8, Pxy = 4, Pyy = -6

 

D = 48 – 16 > 0

Pxx = -8 < 0

 

∴ To maximize profit x = ½,  y = 3.

A)

Px(x, y) = -8x + 4y + 4

Px(1, 3) = -8 + 12 + 4

              = -4 + 2

             = 8

 

B) ∴ To maximize profit x = ½,  y = 3.