The marginal revenue is given by MR = 25 - 5x - 2x^2 and marginal cost MC = 15 - 2x - x^2.
Profit is maximized when MR = MC
=> 25 - 5x - 2x^2 = 15 - 2x - x^2
=> x^2 + 3x - 10 = 0
=> x^2 + 5x - 2x - 10 = 0
=> x(x + 5) - 2(x + 5) = 0
=> (x - 2)(x + 5) = 0
=> x = 2 and x = -5
But as x cannot be negative, the root x = -5 can be eliminated.
Profit is maximized at an output of x = 2.
To find the maximum profit the value of revenue - cost at x = 2 has to be determined but this cannot be done as integration introduces a constant that cannot be determined.
The profit is maximized at an output of 2 though the maximum profit cannot be determined.