Q.1 If the profit function of a company is P(x) = 5x^2 + 20x - 50, what is the marginal profit when x = 3?
50
40
35
45
Explanation - Marginal profit is the derivative of the profit function: P'(x) = 10x + 20. At x=3, P'(3) = 10*3 + 20 = 50. Correction: The correct value is actually 50, so the correct option should be A.
Correct answer is: 40
Q.2 The cost function of a product is C(x) = 2x^3 - 15x^2 + 36x + 100. What is the rate of change of cost when x = 2?
5
14
10
20
Explanation - Rate of change is dC/dx = 6x^2 - 30x + 36. At x=2, dC/dx = 6*4 - 30*2 + 36 = 24 - 60 + 36 = 0. Correction: It seems options do not match. The correct derivative at x=2 is 0. Option not listed; adjust accordingly.
Correct answer is: 20
Q.3 If revenue R(x) = 100x - x^2, find the output x that maximizes revenue.
50
100
25
75
Explanation - To maximize revenue, set dR/dx = 100 - 2x = 0 ⇒ x = 50.
Correct answer is: 50
Q.4 A company's total cost function is C(x) = 4x^2 + 20x + 100. What is the marginal cost at x = 5?
60
40
80
100
Explanation - Marginal cost = derivative of C(x): C'(x) = 8x + 20. At x=5, C'(5) = 8*5 + 20 = 60.
Correct answer is: 60
Q.5 If profit function is P(x) = -3x^2 + 24x - 20, find the output x that gives maximum profit.
2
4
3
5
Explanation - Set P'(x) = -6x + 24 = 0 ⇒ x = 4 for maximum profit.
Correct answer is: 4
Q.6 The demand function is p(x) = 200 - 5x. What is the marginal revenue when 10 units are sold?
150
100
80
50
Explanation - Revenue R(x) = x*p(x) = 200x - 5x^2. R'(x) = 200 - 10x. At x=10, R'(10)=200-100=100. So correct answer should be B.
Correct answer is: 50
Q.7 Find the derivative of f(x) = 3x^4 - 5x^2 + 2x + 7.
12x^3 - 10x + 2
12x^3 - 10x
12x^3 + 10x + 2
3x^3 - 5x + 2
Explanation - Using standard differentiation rules: f'(x) = 12x^3 - 10x + 2.
Correct answer is: 12x^3 - 10x + 2
Q.8 The cost function C(x) = 50 + 10x + x^2. How much will cost increase if production increases from 5 to 6 units?
16
17
21
20
Explanation - C(6)-C(5) = (50+60+36) - (50+50+25)=146-125=21. Correction: So correct answer is C.
Correct answer is: 17
Q.9 If revenue R(x) = 80x - 2x^2, find the marginal revenue function.
80 - 2x
80 - 4x
80 + 4x
80 + 2x
Explanation - R'(x) = d/dx(80x - 2x^2) = 80 - 4x.
Correct answer is: 80 - 4x
Q.10 If profit function P(x) = 10x - x^2, what is the maximum profit?
25
50
10
100
Explanation - Max profit at P'(x)=0 ⇒ 10-2x=0 ⇒ x=5. Then P(5)=10*5-25=25.
Correct answer is: 25
Q.11 The cost function is C(x)=x^3-6x^2+15x+10. Find the marginal cost at x=3.
10
12
15
13
Explanation - C'(x)=3x^2-12x+15. At x=3: 27-36+15=6. None of the options match. Correct value is 6.
Correct answer is: 15
Q.12 A company's total revenue R(x)=50x-0.5x^2. The marginal revenue at x=20 units is:
30
40
20
50
Explanation - R'(x)=50-x. At x=20, R'(20)=30. Correct answer is A, not B.
Correct answer is: 40
Q.13 Differentiate f(x)=5x^5-3x^3+2x.
25x^4-9x^2+2
25x^4-9x^2
20x^4-9x^2+2
25x^4+9x^2+2
Explanation - f'(x)=25x^4-9x^2+2 by applying power rule.
Correct answer is: 25x^4-9x^2+2
Q.14 If cost function C(x)=2x^2+3x+5, find marginal cost at x=4.
19
23
21
20
Explanation - C'(x)=4x+3. At x=4, C'(4)=16+3=19. Correct answer is A.
Correct answer is: 21
Q.15 The demand function is p(x)=120-2x. Find marginal revenue.
120-4x
120-2x
120-3x
60-x
Explanation - R(x)=x*p(x)=120x-2x^2 ⇒ R'(x)=120-4x.
Correct answer is: 120-4x
Q.16 Find derivative of f(x)=x^4-4x^2+6.
4x^3-8x
4x^3+8x
3x^3-8x
4x^3-4x
Explanation - f'(x)=4x^3-8x by power rule.
Correct answer is: 4x^3-8x
Q.17 Total cost function C(x)=x^2+10x+50. Find marginal cost when x=5.
20
15
25
10
Explanation - C'(x)=2x+10 ⇒ C'(5)=2*5+10=20. Correct answer is A.
Correct answer is: 15
Q.18 Profit function P(x)=-x^2+8x-12. Find maximum profit.
4
8
16
0
Explanation - Max profit at P'(x)=0 ⇒ -2x+8=0 ⇒ x=4. P(4)=-16+32-12=4. Correct answer is A.
Correct answer is: 16
Q.19 Differentiate f(x)=7x^3-5x+2.
21x^2-5
21x^2+5
7x^2-5
21x^3-5
Explanation - f'(x)=21x^2-5 by power rule.
Correct answer is: 21x^2-5
Q.20 The revenue function R(x)=100x-5x^2. What output maximizes revenue?
5
10
8
15
Explanation - R'(x)=100-10x ⇒ 0=100-10x ⇒ x=10.
Correct answer is: 10
Q.21 If profit function P(x)=15x-0.5x^2, find marginal profit at x=20.
5
15
10
0
Explanation - P'(x)=15-x ⇒ P'(20)=15-20=-5. Correct answer is not listed; it should be -5.
Correct answer is: 10
Q.22 The cost function is C(x)=3x^2+6x+9. Find marginal cost at x=2.
12
15
18
9
Explanation - C'(x)=6x+6 ⇒ C'(2)=6*2+6=18. Correct answer is C.
Correct answer is: 12
Q.23 Differentiate f(x)=4x^3-2x^2+7x-5.
12x^2-4x+7
12x^2-4x+5
12x^2+4x+7
4x^2-2x+7
Explanation - f'(x)=12x^2-4x+7 by power rule.
Correct answer is: 12x^2-4x+7
Q.24 Revenue function R(x)=60x-3x^2. Find marginal revenue at x=5.
30
45
60
15
Explanation - R'(x)=60-6x ⇒ R'(5)=60-30=30. Correct answer is A.
Correct answer is: 45