For a certain company, the cost for producing x items is 55x+300 and the revenue for selling x items is 95x−0.5x2 .

Question

For a certain company, the cost for producing xx items is 55x+30055x+300 and the revenue for selling xx items is 95x−0.5x295x−0.5x2.

Answer 1

The actual question is missing, but it appears to be the classic: 

    Profit = Revenue – Cost

in which case the question would be:

   What number of items should be produced to maximize profit?

On that assumption, we can write  

    R(x) = 95x - 0.5x^2   and

    C(x) = 55x + 300     and

    P(x) = R(x) - C(x)   or  P(x) = - 0.5x^2 + 40x - 300

To maximize P(x), we take the derivative, and analyze its zeros:

     P’(x) = -x + 40, so the only zero occurs when x = 40

     P’(x) > 0 when x < 40,  P’(x) = 0 when x = 40,  P’(x) < 0 when x > 40

Thus we have a maximum profit when the number of items produced is 40, and that profit is P(40) = $500