Question

The total cost of producing x units is ₹ (x² + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?

Answer 1

Given: Number of units = x

Selling price of each unit = ₹ (180 − x)

∴ Selling price of x unit = ₹ (180 − x) . x

= ₹ (180x − x²)

Also, cost price of x units = ₹ (x² + 60x + 50)

Now, Profit = P = Selling price – Cost price

= (180x − x²) − (x² + 60x + 50) 

= 180x − x² − x² − 60x − 50

∴ P = −2x² + 120x − 50

∴ `(dP)/dx = −4x + 120`

and `(d^2P)/dx^2 = -4`

∴ For maxima and minima,

Consider, `(dP)/dx = 0`

∴ 4x + 120 = 0

∴ −4x = −120

∴ x = 30

For x = 30,

`("d"^2"P")/"dx"^2` = −4 < 0

∴ Total profit is maximum when the number of units = 30.