To reduce shipping distances between the manufacturing facilities and a major co

Question

To reduce shipping distances between the manufacturing facilities and a major consumer, a Korean computer brand, Intel Corp. intends to start production of a new controlling chip for Pentium III microprocessors at their two Asian plants. The cost of producing x1 chips at Chiangmai (Thailand) is

C1 = 0.002x12 + 4x1 + 500,

and the cost of producing x2 chips at Kuala-Lumpur (Malaysia) is

C2 = 0.005x22 + 4x2 + 275.

The Korean computer manufacturer buys them for $150 per chip. Find the quantity that should be produced at each Asian location to maximize the profit if, in accordance with Intel's marketing department, it is described by the expression:

P(x1, x2) = 150(x1 + x2) - C1 - C2.

Get the answer first by analytical means and then confirm it by visualizing the function P(x,y) and determining the extremal value.

Explanation / Answer

Given,

C1 = 0.002x1² + 4x1 + 500,

C2 = 0.005x2² + 4x2 + 275,

The profit is given as

P(x1, x2) = 150(x1 + x2) - C1 - C2.

P(x1, x2) = 150(x1 + x2) - (0.002x1² + 4x1 + 500) - ( 0.005x2² + 4x2 + 275)

P(x1, x2)= -0.002x1²-0.005x2²+146x1+146x2-775

We need to find the maximum value of this function that would give the maximum profit.

We have, P_x1= -0.004x1+146, P_x1x1= -0.004

P_x2= -0.01x2+146, P_x2x2= -.01

P_x1x2=0, P_x2x1=0

Now, the discriminant is

D(x1, x2)=P_x1x1(x1, x2)*P_x2x2(x1, x2)-[P_x1x2]²

The critical points of the function P(x1, x2) is given as

P_x1=0 -0.004x1+146=0 x1=36500

and P_x2=0 -0.01x2+146=0 x2=14600

so, the critical point is (36500, 14600)

Now, at (36500, 14600), the P_x1x1 <0

and D(36500, 14600)=P_x1x1(36500, 14600)*P_x2x2(36500, 14600)-[P_x1x2(36500, 14600)]²

=(-0.004)*(-.01)-(0)²

=0.00004 >0,

Hence, the function P(x1, x2) has maximum value at (36500, 14600)

and the maximum value is

P(36500, 14600)=-0.002(36500)²-0.005(14600)²+146(36500)+146(14600)-775 =3729525

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