Q. For the function f(x) = 3x^2 - 12x + 9, find the vertex. (2021)
-
A.
(2, 3)
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B.
(3, 0)
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C.
(0, 9)
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D.
(1, 6)
Solution
The vertex occurs at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.
Correct Answer: A — (2, 3)
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Q. For the function f(x) = 3x^2 - 12x + 9, find the x-coordinate of the vertex. (2021)
Solution
The vertex x-coordinate is found using -b/(2a) = 12/(2*3) = 2.
Correct Answer: B — 2
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Q. For the function f(x) = x^2 + 2x, find the local maximum. (2022)
Solution
f'(x) = 2x + 2. Setting f'(x) = 0 gives x = -1. f(-1) = 1.
Correct Answer: A — -1
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Q. If f(x) = x^3 - 3x^2 + 4, find the critical points. (2022)
-
A.
1, 2
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B.
0, 3
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C.
2, 4
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D.
1, 3
Solution
f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2.
Correct Answer: A — 1, 2
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Q. If the cost function is C(x) = 3x^2 + 12x + 5, find the minimum cost. (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) = 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
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C.
140
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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. If the revenue function is R(x) = 50x - 0.5x^2, find the number of units that maximizes revenue. (2023)
Solution
Max revenue occurs at x = -b/(2a) = -50/(2*-0.5) = 50.
Correct Answer: A — 25
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Q. What is the derivative of f(x) = 2x^3 - 9x^2 + 12x? (2021)
-
A.
6x^2 - 18x + 12
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B.
6x^2 - 18x
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C.
6x^2 + 18x
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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 area of a triangle with a base of 10 units and height as a function of the base? (2021)
Solution
Area = 0.5 * base * height. Max area occurs when height is 10, giving Area = 0.5 * 10 * 10 = 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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