Question

The total cost C for producing x units is Rs (x² + 60x + 50) and the price is Rs (180 - x) per unit. For how many units the profit is maximum.

Answer 1

Revenue R = x (180 - x)

R = 180x - x²

Profit π(x) = R - C

= (180x - x²) - (x² + 60x + 50)

π(x) = -2x² + 120x - 50

`(dπ)/(dx)` = -4x + 120

Profit π is maximum

If  `(dπ)/(dx)` = 0 and `(d^2y)/(dx^2)` < 0

`(dπ)/(dx)` = 0

-4x + 120 = 0

x = 30

Now, `(d^2π)/(dx^2)` = -4

`(d^2π)/(dx^2)` = -4 < 0at x = 30

Profit is maximum at x = 30 units.