4A company produces and sells $q$ units of a product. The cost function is: $C(q)=2250+2.5 q+0.004 q^{2}$ and the revenue function is:
$R(q)=13 q-0.002 q^{2}$. Which of the following is the production level that maximizes profit?

Select one:
725
575
875
625

Step 1 ：The cost function of the company is given by \(C(q)=2250+2.5 q+0.004 q^{2}\) and the revenue function is given by \(R(q)=13 q-0.002 q^{2}\).

Step 2 ：The profit function is the difference between the revenue function and the cost function, which is \(P(q) = R(q) - C(q) = -0.006q^{2} + 10.5q - 2250\).

Step 3 ：To find the production level that maximizes profit, we need to find the derivative of the profit function and set it equal to zero. This will give us the critical points.

Step 4 ：The derivative of the profit function is \(P'(q) = 10.5 - 0.012q\). Setting this equal to zero gives us the critical point \(q = 875\).

Step 5 ：We need to verify if this critical point is a maximum or a minimum. We can do this by taking the second derivative of the profit function and substituting the critical point into it. If the result is negative, then the critical point is a maximum.

Step 6 ：The second derivative of the profit function is \(P''(q) = -0.012\). Substituting the critical point into it gives us a negative result, which means the critical point is a maximum.

Step 7 ：Final Answer: Therefore, the production level that maximizes profit is \(\boxed{875}\).