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R=(500-2x)x
R=500x-2(x^2)

C=3600+100x+2(x^2)

P=R-C

P=500x-2(x^2)-3600-100x-2(x^2)

P=-4(x^2)+400x-3600
P/4=-x^2+100x-900

Now solve :
P=0
0/4=-x^2+100x-900
0=-x^2+100x-900

Hi,

There are two breakeven points. They are: Making C = R, the following equation is obtained: x^2 - 100.x + 900 = 0. Solving for x, one gets x = 10 units and x = 90 units.

The number of sold units for which the revenue becomes a maximum is obtained making: dR/dx = 500 - 4.x = 0. Solving for x, one gets x = 125 units. At this point the manufacturer is making a loss, because the gross margin (GM = R - C = -3600 + 400.x - 4.x^2) is negative and equal to GM = -16,100 €.

The number of sold units for which the gross margin becomes a maximum is obtained making: dGM/dx = 400 - 8.x = 0. Solving for x, one gets x = 50 units.

See the attached Excel file for a better understanding.