4.8 Using calculus to find maxima and minima in simple problems Quiz: AQA Further Maths, Unit 4

The 20 questions

1. A rectangle has perimeter 40 and width x. What is its area in terms of x?

   • x(20 + x)
   • x(20 - x)
   • x(40 - x)
   • 2x(20 - x)

2. What value of x maximises A = x(20 - x)?

   • 5
   • 15
   • 20
   • 10

3. What is the maximum area of the rectangle with perimeter 40?

   • 50
   • 400
   • 200
   • 100

4. A rectangle has area 16 and sides x and 16/x. Which x minimises its perimeter?

   • 8
   • 2
   • 4
   • 16

5. What is the minimum perimeter of a rectangle of area 16?

   • 32
   • 4
   • 16
   • 8

6. For C = x^2 + 16/x with x > 0, what value of x gives the minimum cost?

   • x = 2
   • x = 4
   • x = 1
   • x = 8

7. For C = x^2 + 16/x, what does d^2C/dx^2 > 0 at x = 2 show?

   • A point of inflection
   • A maximum
   • Only a zero gradient
   • A minimum

8. For x > 0, which value of x minimises x + 1/x?

   • 0
   • 1
   • -1
   • 2

9. What is the minimum value of x + 1/x for x > 0?

   • 2
   • 1
   • 0
   • -2

10. A ball has height h = 12t - 3t^2. At what time is the height greatest?

    • t = 1
    • t = 4
    • t = 6
    • t = 2

11. What is the greatest height of the ball with h = 12t - 3t^2?

    • 36
    • 24
    • 6
    • 12

12. A rectangle has its base on the x-axis and top corners on y = 9 - x^2, giving area A = 2x(9 - x^2). Which x maximises A?

    • sqrt(3)
    • sqrt(6)
    • 3
    • 1.5

13. What is the maximum area for the rectangle described with A = 2x(9 - x^2)?

    • 12 sqrt(3)
    • 6 sqrt(3)
    • 36
    • 18 sqrt(3)

14. What is the minimum value of x + 4/x for x > 0?

    • 2
    • 4
    • 5
    • 3

15. Two positive numbers have sum 20 and product as large as possible. What are they?

    • 10 and 10
    • 8 and 12
    • 4 and 16
    • 5 and 15

16. A cylinder has volume 16 pi, with surface area S = 2 pi r^2 + 32 pi / r. Which r minimises S?

    • 2
    • 1
    • 4
    • 8

17. For the cylinder with S = 2 pi r^2 + 32 pi / r, what is the minimum surface area?

    • 48 pi
    • 16 pi
    • 24 pi
    • 32 pi

18. A rectangle has perimeter 24 and width W. Its area is A = W(12 - W). What width gives the maximum area?

    • 6
    • 4
    • 3
    • 12

19. What is the largest value of y = -x^2 + 4x + 1?

    • 1
    • 9
    • 5
    • 4

20. Profit is P = 100x - 2x^2 for x units. Which x maximises profit?

    • 100
    • 50
    • 12.5
    • 25

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