Question 1

A number x and a second number sum to 20. Their product is P = x(20 − x). Find the maximum product.

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Question 2

Find the coordinates of the stationary point of f(x) = x² − 6x + 5, and state its nature.

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Question 3

Find the coordinates of the minimum point of f(x) = 2x² − 8x + 1.

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Question 4

Find the x-coordinate of the point of inflexion of f(x) = x³ − 9x² + 7x.

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Question 5

Find the stationary points of f(x) = x⁴ − 2x² and classify the one at x = 0.

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Question 6

The curve y = x³ − 12x + 5 has a local maximum. Find its coordinates.

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Question 7

A particle's height is h = 30t − 5t² metres (t in seconds). Find the maximum height reached.

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Question 8

Find the point of inflexion of f(x) = x³ + 3x² − 9x + 1.

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Question 9

Show that f(x) = x³ + 3x has no stationary points.

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Question 10

A rectangle has perimeter 24 cm. With width x, its area is A = x(12 − x). Find the width that maximises the area, and the maximum area.

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Question 11

A farmer encloses a rectangular field against a straight river (no fence needed on the river side) using 60 m of fencing. With width x (the two sides perpendicular to the river), find the dimensions giving the greatest area.

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Question 12

The cost function is C = 2x + 50/x for x > 0. Find the value of x that minimises C and the minimum cost.

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Question 13

A cubic f(x) has f''(x) = 6x − 30. State the x-coordinate of its point of inflexion and the concavity for x < 5.

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Question 14

Find and classify the stationary points of f(x) = x³ − 3x² − 9x + 2.

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Question 15

A curve has stationary points at x = 0 and x = 4. Given f''(x) = 6x − 12, classify each point.

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Question 16

The curve y = x³ + ax has a stationary point at x = 2. Find the value of a.

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Question 17

The curve y = x³ − 3x² + 4 has a stationary point of inflexion? Find any point(s) of inflexion.

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Question 18

Find the coordinates of the point of inflexion of f(x) = x³ − 3x² + 2.

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Question 19

Show that the curve y = x⁴ + 2 has no point of inflexion.

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Question 20

For f(x) = 2x³ − 12x² + 5x − 1, find the point of inflexion.

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Topic 5.8: Stationary Points and Optimisation Questions

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