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प्रश्न
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.
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उत्तर
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.
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