Microeconomics question I'm given this question. I'm not sure if I'm doing it right. Can any one help?
 Given the following short-run cost function: C=0.04q3 - 0.9q2 + 10q + 5
    
    And the constant price of the final commodity of $4 for the firm, determine: 
a. The best level of output for the firm; 
b. The level of profit the firm makes at this level of output;

My answer:
a) Best level of output would be: MR which is 4, since price is 4, perfect competition, MC is the derivative of total cost, which is 0.12q^2-0.18q+10, then equating, 4=0.12q^2-0.18q+10, so q=0.75
b) putting q=0.75 and multiplying price by quantity, level of profit would be 4*0.75
Your first answer would be correct conceptually if this was an interior solution. (The math is incorrect because 2*0.9=1.8 and not 0.18).
However, note that the price falls below the average variable cost. So the firm is better off not producing at all both in the short and long term.
As for your second question: you confuse revenue with profit. Revenue is $p*q$, profit is $p*q-c(q)$
For good measure here are two graphs: one depicts profit for q=0 to 20. Clearly it's negative for the full range 
And here's one depicting average variable costs which you can see never falls below 4.