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Date	May 2022	Marks available	2	Reference code	22M.1.SL.TZ1.7
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	7	Adapted from	N/A

Question

A function, $f$ , has its derivative given by $f' (x) = 3 x^{2} - 12 x + p$ , where $p \in ℝ$ . The following diagram shows part of the graph of $f'$ .

The graph of $f'$ has an axis of symmetry $x = q$ .

The vertex of the graph of $f'$ lies on the $x$ -axis.

The graph of $f$ has a point of inflexion at $x = a$ .

Find the value of $q$ .

[2]

Write down the value of the discriminant of $f'$ .

[1]

b.i.

Hence or otherwise, find the value of $p$ .

[3]

b.ii.

Find the value of the gradient of the graph of $f'$ at $x = 0$ .

[3]

Sketch the graph of $f ″$ , the second derivative of $f$ . Indicate clearly the $x$ -intercept and the $y$ -intercept.

[2]

Write down the value of $a$ .

[1]

e.i.

Find the values of $x$ for which the graph of $f$ is concave-down. Justify your answer.

[2]

e.ii.

Markscheme

EITHER

attempt to use $x = - \frac{b}{2 a}$ (M1)

$q = - \frac{- 12}{2 \times 3}$

attempt to complete the square (M1)

$3 {(x - 2)}^{2} - 12 + p$

attempt to differentiate and equate to $0$ (M1)

$f'' (x) = 6 x - 12 = 0$

THEN

$q = 2$ A1

[2 marks]

discriminant $= 0$ A1

[1 mark]

b.i.

EITHER

attempt to substitute into $b^{2} - 4 a c$ (M1)

${(- 12)}^{2} - 4 \times 3 \times p = 0$ A1

$f' (2) = 0$ (M1)

$- 12 + p = 0$ A1

THEN

$p = 12$ A1

[3 marks]

b.ii.

$f'' (x) = 6 x - 12$ A1

attempt to find $f'' (0)$ (M1)

$= 6 \times 0 - 12$

gradient $= - 12$ A1

[3 marks]

A1A1

Note: Award A1 for line with positive gradient, A1 for correct intercepts.

[2 marks]

$a = 2$ A1

[1 mark]

e.i.

$x < 2$ A1

$f'' (x) < 0$ (for $x < 2$ ) OR the $f''$ is below the $x$ -axis (for $x < 2$ )

OR $f''$ (sign diagram must be labelled $f''$ ) R1

[2 marks]

e.ii.

Examiners report

Candidates did score well on this question. As always, candidates are encouraged to read the questions carefully for key words such as 'value' as opposed to 'expression'. So, if asked for the value of the discriminant, their answer should be a number and not an expression found from $b^{2} - 4 a c$ . As such the value of the discriminant in (b)(i) was often seen in (b)(ii). Please ask students to use a straight edge when sketching a straight line! Overall, the reasoning mark for determining where the graph of f is concave-down, was an improvement on previous years. Sign diagrams were typically well labelled, and the description contained clarity regarding which function was being referred to.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

e.i.

[N/A]

e.ii.

Syllabus sections

Topic 2—Functions » SL 2.6—Quadratic function

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