Q.The total cost function for a firm is $TC = Q^2 - 16Q + 200$. Find the output level that minimises total cost, and confirm it is genuinely a minimum.

Setting the derivative of $TC$ to zero locates the critical output level, and the second derivative confirms whether it is a minimum.

Given: $TC=Q^2-16Q+200$.

Step 1 — Differentiate and set to zero: $\dfrac{d(TC)}{dQ} = 2Q-16$. Setting $2Q-16=0$ gives $Q=8$.

Step 2 — Apply the second derivative test: $\dfrac{d^2(TC)}{dQ^2} = 2$, which is positive, confirming $Q=8$ gives a MINIMUM of total cost (not a maximum) — this makes economic sense, since a cost function's critical point is normally a minimum, unlike a profit function's, which is normally a maximum.

Step 3 — Compute the minimum cost: $TC(8) = 64-128+200=136$. …

Assuming every critical point found from $\frac{d}{dQ}=0$ is automatically a maximum — whether it is a maximum or a minimum depends entirely on the SIGN of the second d …