The utility function of a firm is given by U(q) = -q2 + 80q + 25,300 (dollars). Determine the maximum utility.

1. Start with the utility function:

U(q) = -q^2 + 80q + 25,300

2. Find the first derivative of the utility function to identify the critical points:

\frac{dU}{dq} = -2q + 80

3. Set the first derivative equal to zero and solve for \( q \):

-2q + 80 = 0

-2q = -80

q = 40

4. To confirm that this critical point is a maximum, evaluate the second derivative:

\frac{d^2U}{dq^2} = -2

Since the second derivative is negative, \( U(q) \) has a maximum at \( q = 40 \).

5. Substitute \( q = 40 \) back into the utility function to find the maximum utility:

U(40) = -40^2 + 80 \cdot 40 + 25,300

U(40) = -1600 + 3200 + 25,300

U(40) = 1600 + 25,300

U(40)=26,900

6. Therefore, the maximum utility is:

U(q)=26,900