Q.The marginal cost of a firm is $MC(x) = 6x+4$ (in rupees per unit) and the fixed cost is ₹100. Find the total cost function $TC(x)$, and hence the total cost of producing 10 units.

Total cost is the integral of marginal cost, with the constant of integration fixed by the condition that total cost at zero output equals the fixed cost.

Step 1 — integrate the marginal cost function:

Step 2 — fix the constant using the fixed cost: total cost at zero output equals the fixed cost, so $TC(0)=100$:

So the total cost function is:

Step 3 — evaluate at $x=10$:

$TC(10)=3(10)^{2}+4(10)+100 = 300+40+100 = 440$ …

Setting the constant of integration $C=0$ by default, ignoring the fixed cost entirely — the fixed cost is exactly what determines $C$ here, via $TC(0)=\text{Fixed Cost}$, and dro …