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Stationary Points, Increasing and Decreasing Functions

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 12 marks

The curve y=x^3+kx^2-5 has a stationary point where x=4

k is a constant.

Work out the value of k.

Hint

At a stationary point \dfrac{dy}{dx}=0

Worked solution
  1. \dfrac{dy}{dx}=3x^2+2kx
  2. At a stationary point \dfrac{dy}{dx}=0
  3. Substitute x=4: 48+8k=0
  4. k=-6

Question 21 mark

The curve y=2x+\dfrac{8}{x^2} has one stationary point, where x=2

Which statement about this stationary point is correct?

Select the correct answer.

Hint

Write \dfrac{8}{x^2} as 8x^{-2}, differentiate twice and substitute x=2

Worked solution
  1. y=2x+8x^{-2}
  2. \dfrac{dy}{dx}=2-16x^{-3}, which is 0 when x=2
  3. \dfrac{d^2y}{dx^2}=48x^{-4}=\dfrac{48}{x^4} (the two minuses make a plus)
  4. At x=2: \dfrac{d^2y}{dx^2}=\dfrac{48}{16}=3
  5. Positive second derivative, so the point is a minimum.

Question 3Challenge6 marks

A curve has equation y=x^4-4x^3-8x^2+5

(a)

Work out the x-coordinates of the three stationary points of the curve.

3 marks

Give every value, separated by commas

(b)

Work out the coordinates of the maximum point of the curve.

You must show how you decide which point is the maximum.

2 marks

Write your answer as (x, y)

(c)

For x>0, write down the values of x for which y is decreasing.

Give your answer as an inequality.

1 mark

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Differentiate, set \dfrac{dy}{dx}=0 and take out the common factor 4x first so you don't lose the solution x=0

Worked solution

Part (a)

  1. \dfrac{dy}{dx}=4x^3-12x^2-16x
  2. Set \dfrac{dy}{dx}=0: 4x(x^2-3x-4)=0
  3. 4x(x-4)(x+1)=0
  4. x=-1, x=0 or x=4

Part (b)

  1. \dfrac{d^2y}{dx^2}=12x^2-24x-16
  2. At x=-1: 12+24-16=20>0, minimum
  3. At x=0: -16<0, maximum
  4. At x=4: 192-96-16=80>0, minimum
  5. At x=0, y=5, so the maximum is (0,\,5)

Part (c)

  1. The curve has a maximum at x=0 and a minimum at x=4
  2. So between them it goes down.
  3. 0<x<4

More on this topic: Stationary Points, Increasing and Decreasing Functions worksheet with full solutions

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