Maxima & Minima: Essential Concepts for Competitive Exams

Maxima & Minima

Q. Determine the maximum value of f(x) = -2x^2 + 4x + 1. (2023)

A. 1
B. 2
C. 3
D. 4

Solution

The vertex is at x = -4/(2*(-2)) = 1. The maximum value is f(1) = -2(1)^2 + 4(1) + 1 = 3.

Correct Answer: C — 3

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Q. Determine the maximum value of the function f(x) = -x^2 + 6x - 8. (2022)

A. 0
B. 4
C. 6
D. 8

Solution

The vertex is at x = -6/(2*(-1)) = 3. The maximum value is f(3) = -3^2 + 6*3 - 8 = 1.

Correct Answer: B — 4

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Q. Determine the minimum value of f(x) = x^2 - 6x + 10. (2019)

A. 2
B. 3
C. 4
D. 5

Solution

The minimum occurs at x = -b/(2a) = 6/(2*1) = 3. f(3) = 3^2 - 6(3) + 10 = 3.

Correct Answer: B — 3

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Q. Determine the minimum value of the function f(x) = x^2 - 4x + 6. (2020)

A. 2
B. 3
C. 4
D. 5

Solution

The function is a upward-opening parabola. The minimum occurs at x = -b/(2a) = 4/(2*1) = 2. f(2) = 2^2 - 4(2) + 6 = 2.

Correct Answer: A — 2

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Q. Find the maximum value of f(x) = -2x^2 + 10x - 12. (2023)

A. 2
B. 4
C. 6
D. 8

Solution

The maximum occurs at x = -b/(2a) = 10/(2*2) = 2.5. f(2.5) = -2(2.5^2) + 10(2.5) - 12 = 6.

Correct Answer: D — 8

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Q. Find the maximum value of f(x) = -3x^2 + 12x - 5. (2020)

A. 1
B. 4
C. 7
D. 9

Solution

The maximum occurs at x = -b/(2a) = -12/(-6) = 2. f(2) = -3(2^2) + 12(2) - 5 = 7.

Correct Answer: C — 7

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Q. Find the maximum value of f(x) = -x^2 + 4x + 5. (2021)

A. 5
B. 6
C. 7
D. 8

Solution

The vertex is at x = -4/(2*(-1)) = 2. The maximum value is f(2) = -2^2 + 4*2 + 5 = 7.

Correct Answer: C — 7

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Q. Find the minimum value of f(x) = 4x^2 - 16x + 15. (2023)

A. 1
B. 2
C. 3
D. 4

Solution

The minimum occurs at x = -b/(2a) = 16/(2*4) = 2. f(2) = 4(2^2) - 16(2) + 15 = 1.

Correct Answer: A — 1

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Q. Find the minimum value of f(x) = 4x^2 - 8x + 3. (2022)

A. 0
B. 1
C. 2
D. 3

Solution

The vertex is at x = 8/(2*4) = 1. The minimum value is f(1) = 4(1)^2 - 8(1) + 3 = -1.

Correct Answer: B — 1

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Q. Find the minimum value of f(x) = x^2 + 6x + 10. (2020)

A. 2
B. 4
C. 6
D. 8

Solution

The minimum occurs at x = -b/(2a) = -6/(2*1) = -3. f(-3) = (-3)^2 + 6(-3) + 10 = 1.

Correct Answer: A — 2

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Q. Find the minimum value of the function f(x) = 3x^2 - 12x + 9. (2022)

A. 0
B. 1
C. 3
D. 4

Solution

The minimum occurs at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.

Correct Answer: C — 3

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Q. For the function f(x) = -x^2 + 4x + 1, find the maximum value. (2023)

A. 1
B. 5
C. 7
D. 9

Solution

The maximum occurs at x = -b/(2a) = -4/(-2) = 2. f(2) = -2^2 + 4(2) + 1 = 5.

Correct Answer: B — 5

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Q. For the function f(x) = 3x^2 - 12x + 7, find the coordinates of the minimum point. (2019)

A. (2, -5)
B. (2, -1)
C. (4, 1)
D. (4, -5)

Solution

The vertex is at x = -(-12)/(2*3) = 2. The minimum value is f(2) = 3(2^2) - 12(2) + 7 = -5. Thus, the coordinates are (2, -5).

Correct Answer: A — (2, -5)

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Q. For the function f(x) = x^2 - 6x + 10, what is the minimum value? (2020)

A. 2
B. 3
C. 4
D. 5

Solution

The vertex is at x = 6/2 = 3. The minimum value is f(3) = 3^2 - 6*3 + 10 = 1.

Correct Answer: B — 3

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Q. The minimum value of the function f(x) = x^2 - 4x + 6 occurs at x = ? (2020)

A. 1
B. 2
C. 3
D. 4

Solution

The vertex of the parabola occurs at x = -b/(2a) = 4/2 = 2. The minimum value is f(2) = 2^2 - 4*2 + 6 = 2.

Correct Answer: B — 2

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Q. What is the maximum value of f(x) = -2x^2 + 10x - 12? (2022)

A. 2
B. 4
C. 6
D. 8

Solution

The maximum occurs at x = -b/(2a) = -10/(-4) = 2. f(2) = -2(2^2) + 10(2) - 12 = 6.

Correct Answer: C — 6

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Q. What is the maximum value of f(x) = -3x^2 + 12x - 5? (2019)

A. 1
B. 5
C. 7
D. 9

Solution

The vertex is at x = -12/(2*(-3)) = 2. The maximum value is f(2) = -3(2^2) + 12(2) - 5 = 7.

Correct Answer: C — 7

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Q. What is the minimum value of f(x) = 2x^2 - 8x + 10? (2021)

A. 1
B. 2
C. 3
D. 4

Solution

The minimum occurs at x = -b/(2a) = 8/(2*2) = 2. f(2) = 2(2^2) - 8(2) + 10 = 2.

Correct Answer: A — 1

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Q. What is the minimum value of f(x) = 4x^2 - 16x + 15? (2022)

A. 1
B. 2
C. 3
D. 4

Solution

The minimum occurs at x = -b/(2a) = 16/(2*4) = 2. f(2) = 4(2^2) - 16(2) + 15 = 1.

Correct Answer: B — 2

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Q. What is the minimum value of the function f(x) = 2x^2 + 4x + 1? (2023)

A. 0
B. 1
C. 2
D. 3

Solution

The vertex is at x = -4/(2*2) = -1. The minimum value is f(-1) = 2(-1)^2 + 4(-1) + 1 = -1.

Correct Answer: B — 1

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Q. What is the minimum value of the function f(x) = 3x^2 - 12x + 7? (2019)

A. -5
B. -1
C. 1
D. 5

Solution

The minimum occurs at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 7 = 1.

Correct Answer: B — -1

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Maxima & Minima

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