The total cost function for x units of a commodity is given by $C(x)=\frac{25 x^3}{3}-75 x^2+48 x+34$. The output x at which the marginal cost is minimum is:

3

The correct answer is Option (2) - 3

$C(x) = \frac{25x^3}{3} - 75x^2 + 48x + 34$

$\text{Marginal cost } MC = \frac{dC}{dx}$

$MC = 25x^2 - 150x + 48$

$\frac{d(MC)}{dx} = 50x - 150$

$50x - 150 = 0$

$x = 3$

$\frac{d^2(MC)}{dx^2} = 50 > 0 \Rightarrow \text{minimum}$

The marginal cost is minimum at $x = 3$.