Applications of Derivatives

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Q1. Determine the intervals where f(x) = -x^2 + 4x is concave up. (2023)

Solution:

f''(x) = -2, which is always negative, indicating concave down everywhere.

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Q2. If the cost function is C(x) = 5x^2 + 20x + 100, find the minimum cost. (2020)

Solution:

The minimum cost occurs at x = -b/(2a) = -20/(2*5) = -2. C(-2) = 5(-2)^2 + 20(-2) + 100 = 120.

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Q3. At which point does the function f(x) = -x^3 + 3x^2 + 4 have a local maximum? (2023)

Solution:

Setting f'(x) = -3x^2 + 6x = 0 gives x = 0 and x = 2. f''(2) < 0 indicates a local maximum at (2, 5).

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Q4. 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.

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Q5. A farmer wants to fence a rectangular field with 100 m of fencing. What dimensions will maximize the area? (2022)

Solution:

For maximum area, the rectangle should be a square. Each side = 100/4 = 25 m.

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Q6. Find the local maxima of f(x) = -x^3 + 3x^2 + 1. (2020)

Solution:

f'(x) = -3x^2 + 6x. Setting f'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2. f(2) = 5.

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Q7. Determine the maximum height of the projectile given by h(t) = -16t^2 + 64t + 80. (2023)

Solution:

The maximum height occurs at t = -b/(2a) = -64/(2*-16) = 2. h(2) = -16(2^2) + 64(2) + 80 = 80.

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Q8. Find the maximum area of a triangle with a base of 10 units and height as a function of x. (2022)

Solution:

Area = 1/2 * base * height. Max area occurs when height is maximized, which is 10 units, giving Area = 50.

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Q9. Find the slope of the tangent line to f(x) = 2x^3 - 3x^2 + 4 at x = 1. (2021)

Solution:

f'(x) = 6x^2 - 6. f'(1) = 6(1)^2 - 6 = 0.

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Q10. Determine the maximum value of f(x) = -x^2 + 6x - 8. (2022)

Solution:

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