Applications of Derivatives

Q. At what point does the function f(x) = x^2 - 4x + 5 have a minimum?

A. (2, 1)
B. (1, 2)
C. (0, 5)
D. (4, 1)

Solution

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

Correct Answer: A — (2, 1)

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Q. Determine the slope of the tangent line to the curve y = 2x^3 - 3x^2 + 4 at x = 1.

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

Solution

First, find the derivative: y' = 6x^2 - 6. At x = 1, y' = 6(1)^2 - 6 = 0.

Correct Answer: B — 2

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

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

Solution

The vertex occurs at x = 3. f(3) = -3^2 + 6(3) - 8 = 6.

Correct Answer: C — 6

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Q. Find the point where the function f(x) = x^3 - 3x^2 + 4 has a local minimum.

A. (1, 2)
B. (2, 1)
C. (3, 0)
D. (0, 4)

Solution

First, find f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2. f(2) = 2, which is a local minimum.

Correct Answer: A — (1, 2)

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Q. Find the x-coordinate of the point where the function f(x) = 3x^2 - 12x + 9 has a local maximum.

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

Solution

The vertex occurs at x = -b/(2a) = 12/(2*3) = 2, which is a local maximum.

Correct Answer: B — 2

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Q. For the function f(x) = e^x - x^2, find the critical points.

A. (0, 1)
B. (1, e-1)
C. (2, e-4)
D. (3, e-9)

Solution

Find f'(x) = e^x - 2x. Setting f'(x) = 0 gives e^x = 2x. The critical point is approximately (1, e-1).

Correct Answer: B — (1, e-1)

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Q. What is the derivative of the function f(x) = sin(x) + cos(x)?

A. cos(x) - sin(x)
B. -sin(x) - cos(x)
C. sin(x) + cos(x)
D. -sin(x) + cos(x)

Solution

f'(x) = cos(x) - sin(x).

Correct Answer: A — cos(x) - sin(x)

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Q. What is the inflection point of the function f(x) = x^4 - 4x^3 + 6?

A. (0, 6)
B. (1, 3)
C. (2, 2)
D. (3, 3)

Solution

Find f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(12x - 24) = 0, so x = 0 or x = 2. Check the sign change around x = 2.

Correct Answer: B — (1, 3)

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Q. What is the maximum value of the function f(x) = -2x^2 + 8x - 3?

A. -3
B. 5
C. 8
D. 12

Solution

The function is a downward-opening parabola. The vertex occurs at x = -b/(2a) = -8/(2*-2) = 2. f(2) = -2(2^2) + 8(2) - 3 = 5.

Correct Answer: B — 5

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

A. 0
B. 2
C. 4
D. 8

Solution

The function is a perfect square: f(x) = (x + 2)^2. The minimum value is 0 at x = -2.

Correct Answer: A — 0

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Applications of Derivatives

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