To find the profit at the optimal integer output level $q = 5$, we use the profit function:

$\Pi(q) = -4q^2 + 39q - 350$

Substitute $q = 5$ into the profit function:

$\Pi(5) = -4(5)^2 + 39(5) - 350$ $\Pi(5) = -4(25) + 195 - 350$ $\Pi(5) = -100 + 195 - 350$ $\Pi(5) = 95 - 350$ $\Pi(5) = -255$

The profit at the optimal integer output level of 5 units is -255 dollars.

Do you want more details or have any questions?

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1. What factors can lead to a negative profit even at the optimal output level?
2. How does a negative profit affect business decisions in the short term?
3. What would be the break-even point given the fixed and variable costs?
4. How does the steepness of the marginal cost curve affect profit?
5. What are potential strategies to convert a negative profit into a positive one?
6. How does changing the selling price to $50 per unit affect profit at the same output level?
7. Could reducing fixed costs lead to a positive profit, and by how much?
8. What are the implications of selling fewer units than the optimal output level?

Tip: Always check if external factors like changes in market demand could impact your pricing strategy and profitability.