Question

The total cost function to produce x units of a commodity is 900 +0.030x^2 and total revenue function is 12x. Then how many units should be produced to get maximum profit ? Also obtain maximum profit.​

Answer 1

Given : The total cost function to produce x units of a commodity is 900 +0.030x^2

total revenue function is 12x.

To Find : how many units should be produced to get maximum profit ?   maximum profit.​

Solution:

cost function to produce x unit = 900 +0.030x²

total revenue function = 12x

Profit =  revenue - cost

=> Profit function p(x)  = 12x - (900 +0.030x² )

p   = 12x -  900 - 0.030x²

dp /dx  = 12   -  0.06x

dp/dx  = 0

=>  12   -  0.06x = 0

=> x = 200

d²p /dx² = - 0.06 < 0

Hence maximum profit when   x = 200

200 units should be produced to get maximum profit

Maximum profit =   12(200) -  900 - 0.030(200)²

= 300

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Answer 2

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300