A laptop company has discovered their cost and revenue functions for each day: c (x) = 3x2 - 10x + 200 and r (x) = - 2x2 + 100x + 50. if they want to make a profit, what is the range of laptops per day that they should produce? round to the nearest number which would generate profit.

Given that:

c (x) = 3x2 - 10x + 200

and

r (x) = - 2x2 + 100x + 50

Profit is given by:

P (x) = r (x) - c (x)

P (x) = (-2x2 + 100x + 50) - (3x2 - 10x + 200)

P (x) = - 5x^2+110x-150

thus:

at maximum profit P' (x) = 0

thus:

P' (x) = - 10x+110=0

hence:

x=11

thus the number of units required for one to make profit is 11 units