Q.The Marginal Cost function $MC = 14 - 6x + 4x^2$. Then find the cost function if fixed cost is 12.

Integrate $MC=14-6x+4x^{2}$ and use fixed cost $=C(0)=12$ to fix the constant, giving $C(x)=\tfrac{4}{3}x^{3}-3x^{2}+14x+12$.

Marginal cost is the derivative of total cost, so the total cost function is obtained by integration: $C(x)=\int MC\,dx$.

Step 1 — Integrate the marginal cost:

$C(x)=\int (14-6x+4x^{2})\,dx=14x-3x^{2}+\frac{4}{3}x^{3}+k.$

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