The cost function of a product is given by \( C(x)=\displaystyle\frac{1}{3}x^3-45x^2-900x+36 \), where \( x \) is the number of units produced. In order to minimize the marginal cost, how many units of the product must be produced?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 45
Marginal cost is the derivative of the cost function:
\( \displaystyle MC=C'(x)=x^2-90x-900 \)
MC is minimum where its own derivative is zero:
\( \displaystyle \frac{d}{dx}(MC)=2x-90=0 \)
\( \displaystyle x=45 \)
\( \displaystyle \frac{d^2}{dx^2}(MC)=2\gt 0,\ \text{so MC is minimum at}\ x=45 \)
Shortcut: MC is a quadratic \( x^2-90x-900 \) with a positive leading coefficient; its minimum is at the vertex \( x=-\frac{b}{2a}=\frac{90}{2}=45 \).
Tip: Read what is to be minimised: here it is the marginal cost, so differentiate C(x) twice, not once.