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Grade 9 · Higher · free GCSE maths worksheet

Stationary points and turning points

Edexcel IGCSE

Solving dy/dx = 0 and deciding maximum or minimum from the shape of the graph.

10 original questions, 40 marks. Type your answers below for an instant mark and a score, or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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How to do it

  1. Differentiate to get dy/dx
  2. Stationary points are where the gradient is 0: put dy/dx = 0 and solve for x
  3. Put each x back into the ORIGINAL equation to get y
  4. Max or min? Use the shape: +x² gives a minimum, -x² gives a maximum
  5. Cubic with +x³: the smaller-x stationary point is the max, the larger-x one is the min

Watch out: Substitute x back into the equation of the curve to find y, never into dy/dx (that just gives 0).

Worked example

The curve C has equation y = 2x3 − 3x2 − 12x + 4

Find the coordinates of the two stationary points of C and state whether each is a maximum or a minimum.

dydx=6x2 - 6x - 12

Stationary when the gradient is 0

6x2 - 6x - 12=0
÷6x2 - x - 2=0÷6
(x - 2)(x + 1)=0

x = 2 or x = -1

Put each x into the original equation

x=-1: y = 2(-1) - 3(1) - 12(-1) + 4 = 11
x=2: y = 2(8) - 3(4) - 12(2) + 4 = -16

Positive x3 cubic: max first, then min

(-1, 11) maximum, (2, -16) minimum

The questions

1

The curve y = x2 − 6x + 1 has one turning point.

  1. Use differentiation to find the coordinates of the turning point.
  2. State whether the turning point is a maximum or a minimum.
3 marks
Show answer
a) (3, -8) b) minimum
2

A curve has equation y = 5 + 8x − 2x2

Find the coordinates of the maximum point of the curve.

3 marks
Show answer
(2, 13)
3

The curve C has equation y = x3 − 3x + 6

The point P has coordinates (1, 4).

  1. Show that P is a stationary point of C.
  2. State, with a reason, whether P is a maximum point or a minimum point.
3 marks
Show answer
a) dy/dx = 3x2 - 3 = 0 when x = 1, and y = 1 - 3 + 6 = 4, so P is on C with gradient 0 b) minimum
4

The curve y = x2 + kx + 7, where k is a constant, has its minimum point at x = 4

  1. Find the value of k.
  2. Find the y-coordinate of the minimum point.
4 marks
Show answer
a) k = -8 b) -9
5

A delivery company works out that the cost, £C, of fuel for a journey when a van travels at an average speed of x km/h is given by

C = 0.02x2 − 2x + 60   for 20 ≤ x ≤ 80

  1. Find the speed, in km/h, that makes the fuel cost as small as possible.
  2. Find this smallest fuel cost, in pounds.
4 marks
Show answer
a) 50 b) 10
6

Priya runs a market stall selling strawberries. When she sells x kg of strawberries in a day, her profit, £P, is given by

P = −0.5x2 + 30x − 100

  1. Find the number of kilograms Priya should sell in a day to make the greatest profit.
  2. Find the greatest possible daily profit, in pounds.
4 marks
Show answer
a) 30 b) 350
7

The curve C has equation y = x + 9x   for x > 0

  1. Find the coordinates of the stationary point of C.
  2. State whether this stationary point is a maximum or a minimum.
4 marks
Show answer
a) (3, 6) b) minimum
8

The curve C has equation y = x3 − 6x2 + 9x + 1

  1. Find the x-coordinates of the two stationary points of C.
  2. Write down the coordinates of the maximum point of C.
  3. Write down the coordinates of the minimum point of C.
5 marks
Show answer
a) x = 1 or x = 3 b) (1, 5) c) (3, 1)
9

A farmer has 120 m of fencing. She uses all of it to make a rectangular pen against a long straight stone wall, so the fencing is needed on three sides only.

The two sides at right angles to the wall are each x metres long.

  1. Write down an expression, in terms of x, for the area A m2 of the pen. Give your answer in the form A = px − qx2.
  2. Use differentiation to find the value of x that gives the greatest area.
  3. Find the greatest possible area of the pen, in m2.
5 marks
Show answer
a) A = 120x - 2x2 b) x = 30 c) 1800
10

An open box is made from a square sheet of card of side 24 cm. A square of side x cm is cut from each corner and the sides are folded up.

The volume, V cm3, of the box is given by

V = 4x3 − 96x2 + 576x   for 0 < x < 12

  1. Find an expression for dVdx
  2. Find the value of x that gives the greatest possible volume.
  3. Find the greatest possible volume of the box, in cm3.
5 marks
Show answer
a) 12x2 - 192x + 576 b) x = 4 c) 1024

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Every question here is original and written by Robinson Tuition. This topic is on the Pearson Edexcel International GCSE (4MA1) and is not examined on the UK GCSE; the grade is our estimate against the Edexcel grade descriptors. This is not an exam-board paper.

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