Q. If the cost function is C(x) = 3x^2 + 12x + 5, find the minimum cost. (2020) 2020
Solution
The minimum cost occurs at x = -b/(2a) = -12/(2*3) = -2. C(-2) = 3(-2)^2 + 12(-2) + 5 = 8.
Correct Answer: B — 8
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Q. If the cost function is C(x) = 5x^2 + 20x + 100, find the minimum cost. (2020)
-
A.
100
-
B.
120
-
C.
140
-
D.
160
Solution
The minimum cost occurs at x = -b/(2a) = -20/(2*5) = -2. C(-2) = 5(-2)^2 + 20(-2) + 100 = 120.
Correct Answer: B — 120
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Q. If the revenue function is R(x) = 100x - 2x^2, find the number of units that maximizes revenue. (2021)
Solution
Max revenue occurs at x = -b/(2a) = 100/(2*2) = 25.
Correct Answer: B — 50
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Q. If the revenue function is R(x) = 20x - 0.5x^2, find the quantity that maximizes revenue. (2021)
Solution
R'(x) = 20 - x = 0 gives x = 20. This maximizes revenue.
Correct Answer: B — 20
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Q. What is the derivative of f(x) = 2x^3 - 9x^2 + 12x? (2021)
-
A.
6x^2 - 18x + 12
-
B.
6x^2 - 18x
-
C.
6x^2 + 18x
-
D.
6x^2 - 12
Solution
f'(x) = 6x^2 - 18x + 12.
Correct Answer: A — 6x^2 - 18x + 12
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Q. What is the maximum area of a triangle with a base of 10 cm and height as a function of x? (2020)
Solution
Area = 1/2 * base * height = 1/2 * 10 * x. Max area occurs when x = 10, giving Area = 50.
Correct Answer: B — 50
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Q. What is the maximum area of a triangle with a base of 10 cm and height varying with x? (2021)
Solution
Area = 1/2 * base * height. Max area occurs when height is maximized, thus Area = 1/2 * 10 * 10 = 50.
Correct Answer: B — 50
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Q. What is the maximum area of a triangle with a base of 10 units and height as a function of x? (2020)
Solution
Area = 1/2 * base * height = 5h. Max area occurs when h = 10, giving area = 50.
Correct Answer: B — 50
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Q. What is the maximum height of the projectile modeled by h(t) = -16t^2 + 32t + 48? (2023)
Solution
The maximum height occurs at t = -b/(2a) = 1. h(1) = 64.
Correct Answer: B — 64
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Q. What is the maximum height of the projectile modeled by h(t) = -16t^2 + 64t + 48? (2021)
Solution
The maximum height occurs at t = -b/(2a) = -64/(2*-16) = 2. h(2) = -16(2^2) + 64(2) + 48 = 80.
Correct Answer: B — 64
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Q. What is the maximum profit if the profit function is P(x) = -x^2 + 10x - 16? (2021)
Solution
The maximum profit occurs at x = -b/(2a) = 10/2 = 5. P(5) = -5^2 + 10*5 - 16 = 9.
Correct Answer: C — 8
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Q. What is the maximum value of f(x) = -x^2 + 4x + 1? (2023)
Solution
The vertex occurs at x = 2. f(2) = -2^2 + 4(2) + 1 = 5.
Correct Answer: A — 5
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Q. What is the maximum value of f(x) = -x^2 + 6x - 8? (2023)
Solution
The vertex occurs at x = 3. f(3) = -9 + 18 - 8 = 1.
Correct Answer: C — 6
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Q. What is the minimum distance from the point (3, 4) to the line 2x + 3y - 6 = 0? (2023)
Solution
Using the distance formula from a point to a line, the minimum distance is calculated to be 2 units.
Correct Answer: A — 2
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Q. What is the minimum value of f(x) = 3x^2 - 12x + 12? (2021)
Solution
The vertex occurs at x = 2. f(2) = 3(2^2) - 12(2) + 12 = 0.
Correct Answer: B — 3
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Q. What is the minimum value of f(x) = 3x^2 - 12x + 7? (2022)
Solution
The vertex occurs at x = 2. f(2) = -5.
Correct Answer: A — -5
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Q. What is the minimum value of f(x) = 3x^2 - 12x + 9? (2022)
Solution
The vertex occurs at x = 2. f(2) = 3(2^2) - 12(2) + 9 = 0.
Correct Answer: B — 1
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Q. What is the minimum value of f(x) = x^2 - 4x + 5? (2020)
Solution
The vertex form gives the minimum at x = 2. f(2) = 2.
Correct Answer: A — 1
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Q. What is the minimum value of f(x) = x^2 - 4x + 6? (2022)
Solution
The vertex occurs at x = 2. f(2) = 2.
Correct Answer: B — 3
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Q. What is the minimum value of f(x) = x^2 - 4x + 7? (2023)
Solution
The vertex form gives the minimum at x = 2. f(2) = 2, thus minimum value is 3.
Correct Answer: A — 3
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Q. What is the minimum value of f(x) = x^2 - 6x + 10? (2020)
Solution
The vertex form gives the minimum at x = 3. f(3) = 3^2 - 6(3) + 10 = 1.
Correct Answer: A — 4
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Q. What is the minimum value of the function f(x) = 4x^2 - 16x + 20? (2021)
Solution
The vertex gives the minimum at x = 2. f(2) = 4(2^2) - 16(2) + 20 = 4.
Correct Answer: B — 5
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Q. What is the minimum value of the function f(x) = 4x^2 - 16x + 20? (2021) 2021
Solution
The vertex occurs at x = 2. f(2) = 4(2^2) - 16(2) + 20 = 4.
Correct Answer: B — 5
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Q. What is the point of inflection for the function f(x) = x^3 - 6x^2 + 9x? (2023) 2023
-
A.
(1, 4)
-
B.
(2, 3)
-
C.
(3, 0)
-
D.
(0, 0)
Solution
To find inflection points, set f''(x) = 0. f''(x) = 6x - 12 = 0, x = 2. f(2) = 4.
Correct Answer: A — (1, 4)
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Q. What is the slope of the tangent line to f(x) = x^2 + 2x at x = 1? (2023)
Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 2(1) + 2 = 4.
Correct Answer: B — 3
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Q. What is the slope of the tangent line to the curve y = x^2 - 4x + 5 at x = 3? (2023)
Solution
The slope is given by f'(x) = 2x - 4. At x = 3, f'(3) = 2(3) - 4 = 2.
Correct Answer: C — 2
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Q. What is the slope of the tangent to the curve y = x^2 + 2x at x = 1? (2023)
Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 2(1) + 2 = 4.
Correct Answer: B — 3
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Q. What is the value of f''(x) for f(x) = 4x^3 - 6x^2 + 2 at x = 1? (2022)
Solution
f''(x) = 24x - 12. At x = 1, f''(1) = 24(1) - 12 = 12.
Correct Answer: B — 6
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Q. What is the value of x where f(x) = x^3 - 3x has a local maximum? (2022)
Solution
f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = ±1. f(1) = -2.
Correct Answer: C — 1
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