Q.A firm's total revenue and total cost functions are $R(x) = 40x - x^2$ and $C(x) = x^2 + 8x + 20$. Find the output level that maximises profit, and the maximum profit.

Find $MR$ and $MC$, set them equal to find the candidate output, then confirm it is a maximum using the second-order condition.

Step 1 — Find MR and MC. $MR = R'(x) = 40-2x$. $MC = C'(x) = 2x+8$.

Step 2 — Set $MR=MC$ to find the candidate output.

Step 3 — Check the second-order condition. $P(x)=R(x)-C(x)$, so $P'(x)=MR-MC$ and $P''(x) = MR'(x)-MC'(x) = (-2)-(2) = -4 < 0$. Since $P''(x)<0$, $x=8$ genuinely maximises profit (not minimises it).

Step 4 — Find the maximum profit. $R(8) = 40(8)-(8)^2 = 320-64=256$. $C(8) = (8)^2+8(8)+20 = 64+64+20=148$. So $P(8) = R(8)-C(8) = 256-148=108$. …

Finding the output where $MR=MC$ but skipping the second-order check ($P''(x)<0$) is a genuine gap, not a formality — for some cost/revenue pairs, the point where $MR=MC$ can actually be a minimum of p …