Question P(x) = ใโ5๐ฅใ^2+125๐ฅ+37500 is the total profit function of a company, where x is the production of the company.
Question 1 What will be the production when the profit is maximum? (a) 37500 (b) 12.5 (c) โ12.5 (d) โ37500
Given
P(x) = โ5x2 + 125x + 37500
Differentiating w.r.t x
Pโ(x) = d(ใโ5๐ฅใ^2 + 125๐ฅ + 37500)/๐๐ฅ
Pโ(x) = โ5 ร 2x + 125
Pโ(x) = โ10x + 125
Putting Pโ(x) = 0
โ10x + 125 = 0
โ10x = --125
x = (โ125)/(โ10)
x = 12.5
Finding Pโโ(x)
Pโโ(x) = d(โ10๐ฅ + 125)/๐๐ฅ
Pโโ(x) = โ10
< 0
Since Pโโ(๐) < 0 at ๐ฅ = 12.5
โด ๐ฅ = 12.5 is point of maxima
Thus, P(๐ฅ) is maximum when ๐ = 12.5
So, the correct answer is (b)
Question 2 What will be the maximum profit? (a) Rs 38,28,125 (b) Rs 38281.25 (c) Rs 39,000 (d) None
Putting x = 12.5 in P(x)
P(x) = โ5x2 + 125x + 37500
= โ5(12.5)2 + 125(12.5) + 37500
= โ5(156.25) + 1562.5 + 37500
= โ781.25 + 1562.5 + 37500
= 781.25 + 37500
= Rs 38281.25
So, the correct answer is (b)
Question 3 Check in which interval the profit is strictly increasing . (a) (12.5, โ) (b) for all real numbers (c) for all positive real numbers (d) (0, 12.5)
Profit is strictly increasing where
Pโ(x) > 0
โ10x + 125 > 0
125 > 10x
10x < 125
x < 12.5
So, profit is strictly increasing for x โ (0, 12.5)
So, the correct answer is (d)
Question 4 When the production is 2 units, what will be the profit of the company? (a) 37,500 (b) 37,730 (c) 37,770 (d) None
Putting x = 2 in P(x)
P(x) = โ5x2 + 125x + 37500
= โ5(2)2 + 125(2) + 37500
= โ5(4) + 250 + 37500
= โ20 + 250 + 37500
= 230 + 37500
= Rs 37730
So, the correct answer is (b)
Question 5 What will be production of the company when the profit is Rs 38,250? (a) 15 (b) 30 (c) 2 (d) data is not sufficient to find
Putting P(x) = 38250 in formula of P(x)
P(x) = โ5x2 + 125x + 37500
38250 = โ5x2 + 125x + 37500
5x2 โ 125x โ 37500 + 38250 = 0
5x2 โ 125x + 750 = 0
Dividing both sides by 5
x2 โ 25x + 150 = 0
x2 โ 10x โ 15x + 150 = 0
x(x โ 10) โ 15(x โ 10) = 0
(x โ 10) (x โ 15) = 0
โด x = 10, 15
Since x = 15 is in the options
So, the correct answer is (a)

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.