A metro train from Mehrauli to Gurgoan has capacity to board 900 people. The fare charged (in RS.) is defined by the function $$f=(54-\frac{x}{32})^{2}$$ where ‘x’ is the number of the people per trip.How many people per trip will make the marginal revenue equal to zero?

It is given that the fare charged (in RS.) is defined by the function $$f=(54-\dfrac{x}{32})^{2}$$ where ‘x’ is the number of the people per trip.

Let 'x' be the number of people per trip that will make the marginal revenue equal to zero.

Total revenue generated per trip, g(x) = $$x*(54-\dfrac{x}{32})^{2}$$

When total revenue is maximum, then the marginal revenue will be zero. Therefore, we have to find out the value of 'x' for which g'(x) = 0

$$\Rightarrow$$ $$(54-\dfrac{x}{32})^{2}-\dfrac{2x}{32}*(54-\dfrac{x}{32})=0$$

$$\Rightarrow$$ $$(54-\dfrac{x}{32})^{2}-\dfrac{2x}{32}*(54-\dfrac{x}{32})=0$$

$$\Rightarrow$$ $$(54-\dfrac{x}{32})*(54-\dfrac{3x}{32})=0$$

$$\Rightarrow$$ $$x = 1728, 576$$

It is given that the train has a capacity of 900 only. Hence, x $$\neq$$ 1728 i.e. $$x = 576$$. Therefore, option B is the correct answer.