Discussion
If \(P = 20 - 3q\) is the demand function and \(C = 3 + 2q^{2}\) is cost function, the profit maximizing output is :
Answer: C
The profit maximizing output is 2.
Profit is revenue minus cost, with revenue \(Pq\):
\[ \begin{aligned} \pi &= (20 - 3q)q - (3 + 2q^{2}) \\ &= 20q - 5q^{2} - 3 \\ \frac{d\pi}{dq} &= 20 - 10q = 0 \Rightarrow q = 2 \end{aligned} \]
The second derivative is \(-10 \lt 0\), so this is a maximum. Equivalently, MR = 20 - 6q equals MC = 4q at q = 2. Profit there is 40 - 20 - 3 = 17; at q = 1 it is 12, at q = 3 it is 12 and at q = 4 it is -3, so 1, 3 and 4 all give less.
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