A firm produces and sells x units of a product daily. Its revenue (in rupees) from selling x units is given by R(x)=500x−2x2, while its daily cost (in rupees) is C(x)=50x+200. The daily profit is P(x)=R(x)−C(x).
(i) Obtain an explicit expression for P(x).
(ii) Find all values of x for which the profit is increasing and for which it is decreasing.
(iii) Determine the number of units that must be produced and sold each day to earn the maximum possible profit and calculate that maximum profit.
(iv) Verify the nature of the critical point obtained in part (iii) by using the second derivative.