Grade 12 Maths · Calculus

Find stationary points

Find the turning points of a curve by setting the derivative equal to 0 and solving, then say which are maximums and which are minimums.

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Question 9.2 · May/June 2025 · Paper 1 · National (DBE) · 4 marks

Given:

f (x) = x^{3} - 6 x^{2} + 9 x - 4 = (x - 4) (x - k)^{2}

9.2 Calculate the coordinates of the turning points of

f

.(4)

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More past paper questions: find stationary points

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3. Question 9.2 · May/June 2024 · Paper 1 · National (DBE) · 4 marks
The graph of $f (x) = a x^{3} + b x^{2} + c x - 5$ is drawn below. $E (- 1; 0)$ and $G (5; 0)$ are the $x$ -intercepts of $f$ .
9.2 Calculate the value of x for which f has a local minimum value.(4)
Show the memo (4 ticks)Memo
1. ✓
  Find the derivative of the function $f (x) = x^{3} - 3 x^{2} - 9 x - 5$ $f^{'} (x) = 3 x^{2} - 6 x - 9$
  1 mark · Differentiate using the power rule
2. ✓
  Equate the derivative to zero $3 x^{2} - 6 x - 9 = 0$
  1 mark · Find stationary points
3. ✓
  Factorize the quadratic equation $(x - 3) (x + 1) = 0$
  1 mark · Factorise a trinomial
4. ✓
  Identify the x-value for the local minimum $x = 3$
  1 mark · Find stationary points
4. Question 6.1 · November 2024 · Paper 1 · National (DBE) · 3 marks
In the diagram below, the graphs of $f (x) = - x^{2} + 4 x + 5$ and $g$ , a straight line, are drawn. $C (3; 8)$ is a point of intersection of $f$ and $g$ . $E H$ is drawn parallel to the y-axis, with $E$ a point on $f$ and $H$ a point on $g$ .
6.1 Calculate the coordinates of $B$ , the turning point of $f$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Calculate x-coordinate of turning point $x = \frac{- b}{2 a} = \frac{- 4}{2 ( - 1 )}$
  $x = 2$
  1 mark · Find turning point of a parabola
2. ✓
  Calculate y-coordinate $y = - (2)^{2} + 4 (2) + 5$
  $y = 9$
  1 mark · Find turning point of a parabola
3. ✓
  Write final coordinates $B (2; 9)$
  1 mark · Find turning point of a parabola
5. Question 10.1 · November 2024 · Paper 1 · National (DBE) · 3 marks
A cyclist rode from town P and stopped at town T. The speed (in km/h) at which this cyclist rode, is represented by the equation $s^{'} (t) = - 3 t^{2} + 18 t$ . NOTE: Speed is the rate of change in distance with respect to time.
10.1 Calculate the maximum speed that the cyclist reached on this ride.(3)
Show the memo (2 ticks)Memo
1. ✓
  Determine time of max speedMax occurs at vertex of parabola $s^{'} (t)$ .
  $t = \frac{- b}{2 a} = \frac{- 18}{2 ( - 3 )} = 3$
  1 mark · Find turning point of a parabola
2. ✓
  Calculate max speed $s^{'} (3) = - 3 (3)^{2} + 18 (3) = 27$ km/h
  2 marks · Find turning point of a parabola
6. Question 8.3 · September 2024 · Paper 1 · Provincial · 6 marks
8.3 A function $g (x) = a x^{2} + \frac{b}{x}$ has a minimum value at $x = 4$ . The function value at $x = 4$ is 96. Calculate the values of a and b.(6)
Show the memo (6 ticks)Memo
1. ✓
  Derivative is 0 at the minimumMinimum at $x = 4$ , so $g^{'} (4) = 0$
  1 mark · Find stationary points
2. ✓
  Derivative $g (x) = a x^{2} + b x^{- 1}$ $g^{'} (x) = 2 a x - b x^{- 2}$
  1 mark · Differentiate using the power rule
3. ✓
  Equation 1 $2 a (4) - b (4)^{- 2} = 0$ $8 a - \frac{b}{16} = 0$ $128 a - b = 0$ ...(1)
  1 mark · Find stationary points
4. ✓
  Equation 2 $g (4) = 96$ : $16 a + \frac{b}{4} = 96$ $64 a + b = 384$ ...(2)
  1 mark · Substitute into a formula
5. ✓
  Value of a(1) + (2): $192 a = 384$ $a = 2$
  1 mark · Solve simultaneous equations by elimination
6. ✓
  Value of bFrom (1): $b = 128 (2) = 256$
  1 mark · Solve simultaneous equations by substitution

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