Question

A television manufacturer finds that the total cost for the production and marketing of x number of television sets is C(x) = 300x² + 4200x + 13500. If each product is sold for ₹ 8,400. show that the profit of the company is increasing.

Answer 1

C(x) = 300x² + 4200x + 13,500

Selling price of one product = ₹ 8,400

Selling price of x numbers of products = 8400x

Profit, P = Selling price – Cost price

= 8400x – (300x² + 4200x + 13500)

= 8400x – 300x² – 4200x – 13500

P = - 300x² + 4200x – 13500

Differentiating with respect to x we get

P'(x) = `"dP"/"dx"` = -600x + 4200

`"dP"/"dx"` = 0 gives -600x + 4200 = 0

- 600x = - 4200

x = 7

The point x = 7 divide the real numbers into the intervals (0, 7), (7, ∞). Here x cannot be negative.
Now P'(x) = – 600x + 4200

Take x = 2 in (0, 7)

P'(2) = - 600 × 2 + 4200

= - 1200 + 4200

= 3000, positive

∴ P'(x) is increasing in (0, 7) the profit of the company increasing when each product is sold for ₹ 8,400.