Question 2 - Case Based Questions (MCQ) - Chapter 6 Class 12 Application of Derivatives
Last updated at April 16, 2024 by Teachoo
P(x) = -5x
2
+125x + 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) –12.5
(d) –37500
Question 2
What will be the maximum profit?
(a) Rs 38,28,125
(b) Rs 38281.25
(c) Rs 39,000
(d) None
Question 3
Check in which interval the profit is strictly increasing .
(a) (12.5, ∞)
(b) for all real numbers
(c) for all positive real numbers
(d) (0, 12.5)
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
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
Transcript
Question P(x) = 〖−5𝑥〗^2+125𝑥+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) –12.5 (d) –37500
Given
P(x) = −5x2 + 125x + 37500
Differentiating w.r.t x
P’(x) = d(〖−5𝑥〗^2 + 125𝑥 + 37500)/𝑑𝑥
P’(x) = −5 × 2x + 125
P’(x) = −10x + 125
Putting P’(x) = 0
−10x + 125 = 0
−10x = --125
x = (−125)/(−10)
x = 12.5
Finding P’’(x)
P’’(x) = d(−10𝑥 + 125)/𝑑𝑥
P’’(x) = −10
< 0
Since P’’(𝒙) < 0 at 𝑥 = 12.5
∴ 𝑥 = 12.5 is point of maxima
Thus, P(𝑥) is maximum when 𝒙 = 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) = −5x2 + 125x + 37500
= −5(12.5)2 + 125(12.5) + 37500
= −5(156.25) + 1562.5 + 37500
= −781.25 + 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, ∞) (b) for all real numbers (c) for all positive real numbers (d) (0, 12.5)
Profit is strictly increasing where
P’(x) > 0
−10x + 125 > 0
125 > 10x
10x < 125
x < 12.5
So, profit is strictly increasing for x ∈ (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) = −5x2 + 125x + 37500
= −5(2)2 + 125(2) + 37500
= −5(4) + 250 + 37500
= −20 + 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) = −5x2 + 125x + 37500
38250 = −5x2 + 125x + 37500
5x2 − 125x − 37500 + 38250 = 0
5x2 − 125x + 750 = 0
Dividing both sides by 5
x2 − 25x + 150 = 0
x2 − 10x − 15x + 150 = 0
x(x − 10) − 15(x − 10) = 0
(x − 10) (x − 15) = 0
∴ x = 10, 15
Since x = 15 is in the options
So, the correct answer is (a)
Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.
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