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Date	May 2021	Marks available	3	Reference code	21M.3.AHL.TZ1.1
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 1
Command term	Sketch	Question number	1	Adapted from	N/A

Question

This question asks you to explore the behaviour and key features of cubic polynomials of the form $x^{3} - 3 c x + d$ .

Consider the function $f (x) = x^{3} - 3 c x + 2$ for $x \in ℝ$ and where $c$ is a parameter, $c \in ℝ$ .

The graphs of $y = f (x)$ for $c = - 1$ and $c = 0$ are shown in the following diagrams.

$c = - 1$ $c = 0$

On separate axes, sketch the graph of $y = f (x)$ showing the value of the $y$ -intercept and the coordinates of any points with zero gradient, for

Hence, or otherwise, find the set of values of $c$ such that the graph of $y = f (x)$ has

Given that the graph of $y = f (x)$ has one local maximum point and one local minimum point, show that

Hence, for $c > 0$ , find the set of values of $c$ such that the graph of $y = f (x)$ has

$c = 1$ .

[3]

a.i.

$c = 2$ .

[3]

a.ii.

Write down an expression for $f' (x)$ .

[1]

a point of inflexion with zero gradient.

[1]

c.i.

one local maximum point and one local minimum point.

[2]

c.ii.

no points where the gradient is equal to zero.

[1]

c.iii.

the $y$ -coordinate of the local maximum point is $2 c^{\frac{3}{2}} + 2$ .

[3]

d.i.

the $y$ -coordinate of the local minimum point is $- 2 c^{\frac{3}{2}} + 2$ .

[1]

d.ii.

exactly one $x$ -axis intercept.

[2]

e.i.

exactly two $x$ -axis intercepts.

[2]

e.ii.

exactly three $x$ -axis intercepts.

[2]

e.iii.

Consider the function $g (x) = x^{3} - 3 c x + d$ for $x \in ℝ$ and where $c, d \in ℝ$ .

Find all conditions on $c$ and $d$ such that the graph of $y = g (x)$ has exactly one $x$ -axis intercept, explaining your reasoning.

[6]

Markscheme

$c = 1$ : positive cubic with correct $y$ -intercept labelled A1

local maximum point correctly labelled A1

local minimum point correctly labelled A1

[3 marks]

a.i.

$c = 2$ : positive cubic with correct $y$ -intercept labelled A1

local maximum point correctly labelled A1

local minimum point correctly labelled A1

Note: Accept the following exact answers:
Local maximum point coordinates $(- \sqrt{2}, 2 + 4 \sqrt{2})$ .
Local minimum point coordinates $(\sqrt{2}, 2 - 4 \sqrt{2})$ .

[3 marks]

a.ii.

$f' (x) = 3 x^{2} - 3 c$ A1

Note: Accept $3 x^{2} - 3 c$ (an expression).

[1 mark]

$c = 0$ A1

[1 mark]

c.i.

considers the number of solutions to their $f' (x) = 0$ (M1)

$3 x^{2} - 3 c = 0$

$c > 0$ A1

[2 marks]

c.ii.

$c < 0$ A1

Note: The (M1) in part (c)(ii) can be awarded for work shown in either (ii) or (iii).

[1 mark]

c.iii.

attempts to solve their $f' (x) = 0$ for $x$ (M1)

$x \pm \sqrt{c}$ (A1)

Note: Award (A1) if either $x = - \sqrt{c}$ or $x = \sqrt{c}$ is subsequently considered.
Award the above (M1)(A1) if this work is seen in part (c).

correctly evaluates $f (- \sqrt{c})$ A1

$f (- \sqrt{c}) = - c^{\frac{3}{2}} + 3 c^{\frac{3}{2}} + 2 (= - c \sqrt{c} + 3 c \sqrt{c} + 2)$

the $y$ -coordinate of the local maximum point is $2 c^{\frac{3}{2}} + 2$ AG

[3 marks]

d.i.

correctly evaluates $f (\sqrt{c})$ A1

$f (\sqrt{c}) = c^{\frac{3}{2}} - 3 c^{\frac{3}{2}} + 2 (= c \sqrt{c} - 3 c \sqrt{c} + 2)$

the $y$ -coordinate of the local minimum point is $- 2 c^{\frac{3}{2}} + 2$ AG

[1 mark]

d.ii.

the graph of $y = f (x)$ will have one $x$ -axis intercept if

EITHER

$- 2 c^{\frac{3}{2}} + 2 > 0$ (or equivalent reasoning) R1

the minimum point is above the $x$ -axis R1

Note: Award R1 for a rigorous approach that does not (only) refer to sketched graphs.

THEN

$0 < c < 1$ A1

Note: Condone $c < 1$ . The A1 is independent of the R1.

[2 marks]

e.i.

the graph of $y = f (x)$ will have two $x$ -axis intercepts if

EITHER

$- 2 c^{\frac{3}{2}} + 2 = 0$ (or equivalent reasoning) (M1)

evidence from the graph in part(a)(i) (M1)

THEN

$c = 1$ A1

[2 marks]

e.ii.

the graph of $y = f (x)$ will have three $x$ -axis intercepts if

EITHER

$- 2 c^{\frac{3}{2}} + 2 < 0$ (or equivalent reasoning) (M1)

reasoning from the results in both parts (e)(i) and (e)(ii) (M1)

THEN

$c > 1$ A1

[2 marks]

e.iii.

case 1:

$c \leq 0$ (independent of the value of $d$ ) A1

EITHER

$g' (x) = 0$ does not have two solutions (has no solutions or $1$ solution) R1

$\Rightarrow g' (x) \geq 0$ for $x \in^{~}$ R1

the graph of $y = f (x)$ has no local maximum or local minimum points, hence any vertical translation of this graph ( $y = g (x)$ ) will also have no local maximum or local minimum points R1

THEN

therefore there is only one $x$ -axis intercept AG

Note: Award at most A0R1 if only $c < 0$ is considered.

case 2

$c > 0$

$(- \sqrt{c}, 2 c^{\frac{3}{2}} + d)$ is a local maximum point and $(\sqrt{c}, - 2 c^{\frac{3}{2}} + d)$ is a local minimum point (A1)

Note: Award (A1) for a correct $y$ -coordinate seen for either the maximum or the minimum.

considers the positions of the local maximum point and/or the local minimum point (M1)

EITHER
considers both points above the $x$ -axis or both points below the $x$ -axis

OR

considers either the local minimum point only above the $x$ -axis OR the local maximum point only below the $x$ -axis

THEN

$d > 2 c^{\frac{3}{2}}$ (both points above the $x$ -axis) A1

$d < - 2 c^{\frac{3}{2}}$ (both points above the $x$ -axis) A1

Note: Award at most (A1)(M1)A0A0 for case 2 if $c > 0$ is not clearly stated.

[6 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

c.iii.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

e.i.

[N/A]

e.ii.

[N/A]

e.iii.

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.3—Graphing

Show 53 related questions

22M.3.AHL.TZ1.2h.ii:
For $a = r$ and $b > 0$ , state in terms of $r$ , the coordinates of points $P$ and $A$ .
22M.3.AHL.TZ1.2h.i:
Sketch the curve $y = (x - r) (x^{2} - 2 a x + a^{2} + b^{2})$ for $a = r = 1$ and $b = 2$ .
22M.1.SL.TZ1.7d:
Sketch the graph of $f ″$ , the second derivative of $f$ . Indicate clearly the $x$ -intercept and the $y$ -intercept.
22M.1.SL.TZ2.4a.ii:
Write down the equation of the horizontal asymptote.
22M.1.SL.TZ2.4a.i:
Write down the equation of the vertical asymptote.
22M.1.SL.TZ2.4b:
On the set of axes below, sketch the graph of $y = f (x)$ .

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.
22M.1.AHL.TZ2.3b:
On the set of axes below, sketch the graph of $y = f (x)$ .

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.
18N.2.AHL.TZ0.H_9c.i:
sketch the graph of $y = f\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
19M.2.AHL.TZ2.H_4a:
Sketch the graphs $y = {\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x$ and $y = 1 + {\text{cos}}\,x$ on the following axes for 0 < $x$ ≤ 9.
18M.1.AHL.TZ2.H_2a:
Sketch the graphs of $y = \frac{x}{2} + 1$ and $y = \left| {x - 2} \right|$ on the following axes.
18N.2.SL.TZ0.T_4a:
Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.
18N.2.SL.TZ0.T_4b.iii:
Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).

Give your answer in the form y = mx + c.
18N.2.SL.TZ0.T_4c:
Sketch the graph of the function g (x) = 10x + 40 on the same axes.
19M.2.SL.TZ2.T_5b:
Write down the $y$ -intercept of the graph of $y = f\left( x \right)$ .
21N.3.AHL.TZ0.1f:
Sketch the graph of $x^{2} - y^{2} = 1$ , stating the coordinates of any axis intercepts and the equation of each asymptote.
19N.1.SL.TZ0.S_10c:
The line ${L_2}$ is tangent to the graph of $g$ at ${\text{A}}$ and has equation $y = \left( {{\text{ln}}\,p} \right)x + q + 1$ .

The line ${L_2}$ passes through the point $\left( { - 2{\text{, }} - 2} \right)$ .

The gradient of the normal to $g$ at ${\text{A}}$ is $\frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}$ .

Find the equation of ${L_1}$ in terms of $x$ .
19N.1.SL.TZ0.S_10a:
Write down the coordinates of ${\text{B}}$ .
18N.2.AHL.TZ0.H_9c.ii:
sketch the graph of $y = {f^{ - 1}}\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
19M.2.SL.TZ2.T_5c:
Sketch the graph of $y = f\left( x \right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.
18M.2.AHL.TZ2.H_10a.i:
Sketch the graph of $y = f\left( x \right)$ for $- \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}$ .
18M.1.AHL.TZ1.H_9b.ii:
The coordinates of B can be expressed in the form B $\left( {{2^a},\,b \times {2^{ - 3a}}} \right)$ where a, b $\in \mathbb{Q}$ . Find the value of a and the value of b.
22M.3.AHL.TZ2.1a.i:
$y^{2} = x^{3}, x \geq 0$
22M.3.AHL.TZ2.1a.ii:
$y^{2} = x^{3} + 1, x \geq - 1$
22M.1.SL.TZ2.4c:
Hence, solve the inequality $0 < \frac{2 x - 1}{x + 1} < 2$ .
22M.2.AHL.TZ2.10a.ii:
Plant $A$ correct to three significant figures.
21M.1.SL.TZ1.8e:
Sketch the graph of $f$ , showing clearly the value of the $x$ -intercept and the approximate position of point $P$ .
18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .
17N.2.SL.TZ0.T_5b.ii:
Find $f’(x)$ .
21M.3.AHL.TZ2.1b:
Use your graphic display calculator to explore the graph of $y = f_{n} (x)$ for

• the odd values $n = 3$ and $n = 5$ ;

• the even values $n = 2$ and $n = 4$ .

Hence, copy and complete the following table.
21M.3.AHL.TZ1.1a.ii:
$c = 2$ .
21N.2.SL.TZ0.2b:
Sketch the graph of $f$ on the following grid.
17N.1.AHL.TZ0.H_6a:
Sketch the graph of $y = \frac{{1 - 3x}}{{x - 2}}$ , showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.
19N.1.SL.TZ0.S_10b:
Given that $f'\left( a \right) = \frac{1}{{{\text{ln}}\,p}}$ , find the equation of ${L_1}$ in terms of $x$ , $p$ and $q$ .
19N.1.SL.TZ0.S_3b:
The graph of $f\left( x \right) = {x^2}$ is transformed to obtain the graph of $g$ .

Describe this transformation.
21N.3.AHL.TZ0.1b:
Show that ${(f (t))}^{2} + {(g (t))}^{2} = f (2 t)$ .
21N.3.AHL.TZ0.1c.i:
$f (i u)$ .
21N.3.AHL.TZ0.1g:
The hyperbola with equation $x^{2} - y^{2} = 1$ can be rotated to coincide with the curve defined by $x y = k, k \in ℝ$ .

Find the possible values of $k$ .
22M.2.AHL.TZ1.10a:
Sketch the curve $y = f (x)$ , clearly indicating the coordinates of the endpoints.
22M.1.AHL.TZ2.11a:
Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
22M.2.AHL.TZ2.10a.i:
Plant $B$ .
21N.3.AHL.TZ0.1a:
Verify that $u = f (t)$ satisfies the differential equation $\frac{d^{2} u}{d t^{2}} = u$ .
21N.3.AHL.TZ0.1e:
Show that ${(f (t))}^{2} - {(g (t))}^{2} = {(f (i u))}^{2} - {(g (i u))}^{2}$ .
17N.2.SL.TZ0.T_5b.i:
Expand the expression for $f(x)$ .
21N.2.SL.TZ0.2a:
Find the values of $x$ for which $f (x) = 0$ .
19N.1.SL.TZ0.S_3a:
Find the value of $b$ .
21M.3.AHL.TZ2.1a:
Sketch the graph of $y = f_{1} (x)$ , stating the values of any axes intercepts and the coordinates of any local maximum or minimum points.
18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .
19M.2.SL.TZ2.T_5h:
Determine the range of $f\left( x \right)$ for $p$ ≤ $x$ ≤ $q$ .
17N.2.SL.TZ0.T_5e:
Write down the coordinates of the point of intersection.
21N.3.AHL.TZ0.1c.ii:
$g (i u)$ .
21N.3.AHL.TZ0.1d:
Hence find, and simplify, an expression for ${(f (i u))}^{2} + {(g (i u))}^{2}$ .
18M.2.SL.TZ1.T_4e:
Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.
17N.2.SL.TZ0.T_5d:
Draw the graph of $f$ for $- 3 \leqslant x \leqslant 3$ and $- 40 \leqslant y \leqslant 20$ . Use a scale of 2 cm to represent 1 unit on the $x$ -axis and 1 cm to represent 5 units on the $y$ -axis.

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Topic 2—Functions

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