A soft-drink vendor at a popular beach analyzes his sales records and finds that if he sells x cans of soda pop in one day, his profit (in dollars) is given by P (x) = - 0.001x2 + 3x - 1850.

First derivative of the total profit function is

dP/dx = d/dx (-0.001x^2 + 4x - 1850)

= - 0.002x + 4

When dP/dx is taken equal to 0

- 0.002x + 4 =

x = 4/0.002 = 2000

Second derivative is

d^2P/dx^2 = d/dx (dP/dx)

= d/dx (-0.002x + 4)

= - 0.002

Since the second derivative is - ve, it can be said the profit function has a maximum at x = 2000

Profit will be maximum when 2000 cans are sold

1) Maximum profit = P (x=2000) = - 0.001 (2000) ^2 + 4*2000 - 1850

= - 4000 + 8000 - 1850

= 2150

2) He must sell 2000 cans for getting maximum profit.