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Date	May 2021	Marks available	6	Reference code	21M.3.AHL.TZ1.1
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 1
Command term	Find	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 2.4—Key features of graphs, intersections using technology

Show 100 related questions

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.TZ2.3a.i:
Write down the equation of the vertical asymptote.
22M.2.SL.TZ2.8a.ii:
Plant $A$ correct to three significant figures.
18M.2.SL.TZ2.T_6a:
Sketch the curve for −1 < x < 3 and −2 < y < 12.
16N.2.AHL.TZ0.H_5c:
Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .
17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation $f(x) = 5$ .
17M.2.AHL.TZ1.H_12c:
Explain why $f$ is an even function.
19M.2.SL.TZ1.T_4a:
Find the value of $f\left( x \right)$ when $x = \frac{1}{2}$ .
17M.2.AHL.TZ1.H_12b:
Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
19N.2.SL.TZ0.S_7b:
Find the difference between the greatest possible expected value and the least possible expected value.
22M.3.AHL.TZ2.1b.ii:
By considering each curve from part (a), identify two key features that would distinguish one curve from the other.
22M.3.AHL.TZ2.1f.ii:
Hence, find the coordinates of the rational point $Q$ where this tangent intersects $C$ , expressing each coordinate as a fraction.
22M.1.SL.TZ2.4c:
Hence, solve the inequality $0 < \frac{2 x - 1}{x + 1} < 2$ .
22M.1.AHL.TZ2.3a.ii:
Write down the equation of the horizontal asymptote.
22M.1.AHL.TZ2.3c:
Hence, solve the inequality $0 < \frac{2 x - 1}{x + 1} < 2$ .
22M.2.SL.TZ1.8c.i:
Find the value of $p$ and the value of $q$ .
EXN.1.AHL.TZ0.11a:
Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.
22M.2.AHL.TZ2.10a.ii:
Plant $A$ correct to three significant figures.
22M.2.AHL.TZ2.10b:
Find the values of $t$ when $h_{A} (t) = h_{B} (t)$ .
21M.2.AHL.TZ1.7b:
Find the value of $k$ .
20N.2.SL.TZ0.S_1b:
Solve $f (x) = 0$ .
17M.2.SL.TZ2.S_10c:
Jill plays the game nine times. Find the probability that she wins exactly two prizes.
21M.2.AHL.TZ1.11d:
Sketch the graph of $y = f (x)$ for $- 3 \leq x \leq 3$ , showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.
16N.2.AHL.TZ0.H_5b:
State the range of $f$ .
21M.2.SL.TZ2.2a:
Sketch the graph of $f$ on the grid below.
17M.2.AHL.TZ1.H_12d:
Explain why the inverse function ${f^{ - 1}}$ does not exist.
17M.2.AHL.TZ1.H_12g.i:
Hence, show that there are no solutions to $g'(x) = 0$ ;
20N.2.SL.TZ0.S_1c:
The graph of $f$ has a local minimum at point $A$ .

Find the coordinates of $A$ .
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
21M.2.AHL.TZ2.11d.ii:
State the cross-sectional radius of the bowl at this point.
21M.2.AHL.TZ2.11d.i:
By sketching the graph of a suitable derivative of $f$ , find where the cross-sectional radius of the bowl is decreasing most rapidly.
21M.2.AHL.TZ2.12f:
Sketch the graph of $y = g^{- 1} (x)$ , clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.
20N.1.SL.TZ0.T_4a.i:
Write down the zero of $f (x)$ .
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
EXN.2.SL.TZ0.9e:
Find the time taken for the water to have a temperature of $50 ° C$ . Give your answer correct to the nearest second.
16N.2.AHL.TZ0.H_5a:
Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.
17M.2.AHL.TZ1.H_12e:
Find the inverse function ${g^{ - 1}}$ and state its domain.
20N.2.SL.TZ0.S_1a:
Find $f (1)$ .
EXN.2.SL.TZ0.6a:
Find the $x$ -coordinates of the points of intersection of the two curves.
EXN.2.SL.TZ0.9d:
Sketch the graph of $T$ versus $t$ , clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.
21N.2.SL.TZ0.5b:
Hence determine the value of $θ$ .
21N.2.SL.TZ0.7b.iii:
Write down the minimum value of $T$ .
21N.2.SL.TZ0.7c.ii:
Write down the minimum total cost for this journey.
21N.2.SL.TZ0.8a:
Show that $b = \frac{π}{6}$ .
21N.2.SL.TZ0.8c:
Find the value of $d$ .
21N.2.SL.TZ0.8d:
Find the smallest possible value of $c$ .
22M.2.AHL.TZ1.10a:
Sketch the curve $y = f (x)$ , clearly indicating the coordinates of the endpoints.
22M.2.AHL.TZ2.10a.i:
Plant $B$ .
22M.3.AHL.TZ2.1g:
The point $S (- 1, 1)$ also lies on $C$ . The line $[QS]$ intersects $C$ at a further point. Determine the coordinates of this point.
22M.3.AHL.TZ2.1c:
By varying the value of $b$ , suggest two key features common to these curves.
22M.1.SL.TZ2.8a:
Find the coordinates of $B$ .
22M.2.SL.TZ2.8a.i:
Plant $B$ .
21N.2.SL.TZ0.7b.i:
Find an expression, in terms of $x$ for the travel time $T$ , from $A$ to $B$ , passing through $D$ .
21N.2.SL.TZ0.8b:
Find the value of $a$ .
21N.2.AHL.TZ0.9a:
Show that $b = \frac{π}{6}$ .
21N.2.AHL.TZ0.9b:
Find the value of $a$ .
21N.2.AHL.TZ0.9d:
Find the smallest possible value of $c$ .
19M.2.SL.TZ2.T_6a:
Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places.
19M.2.SL.TZ2.T_6c:
After $m$ complete months Tommaso will, for the first time, have enough money in his account to buy the bicycle.

Find the value of $m$ .
17M.2.SL.TZ2.T_6d.i:
Write down the $x$ -coordinates of these two points;
20N.1.SL.TZ0.T_4b:
Consider the function $g (x) = 3 - x$ .

Solve $f (x) = g (x)$ .
21N.2.SL.TZ0.5a:
Given that the areas of the two shaded regions are equal, show that $θ = 2 \sin θ$ .
21N.2.SL.TZ0.7a.ii:
the boat travels directly to $B$ .
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
16N.2.SL.TZ0.T_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
19N.2.SL.TZ0.S_7a:
Show that $b = 0.3 - a$ .
21M.2.SL.TZ1.9a:
Find the $x$ -coordinate of $P$ .
17M.2.AHL.TZ1.H_12g.ii:
Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .
21M.2.AHL.TZ1.12d:
Find the value of $k$ , giving your answer correct to four significant figures.
EXN.2.SL.TZ0.9f:
The model for the temperature of the water can also be expressed in the form $T = T_{0} a^{\frac{t}{10}}$ for $t \geq 0$ and $a$ is a positive constant.

Find the exact value of $a$ .
21M.2.AHL.TZ1.11f:
By considering the graph of $y = g (x) - f (x)$ , or otherwise, solve $f (x) < g (x)$ for $x \in ℝ$ .
21M.2.SL.TZ2.2b:
Find the value of $x$ for which $f' (x) = 0$ .
21N.2.AHL.TZ0.9e:
Find the height of the water at $12 : 00$ .
19M.2.SL.TZ1.T_4d:
Find the distance from the centre of Orangeton to the point at which the road meets the highway.
19M.2.SL.TZ2.T_6b:
Find the difference between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places.
19M.2.SL.TZ1.T_4c:
Find the coordinates of the bridges relative to the centre of Orangeton.
20N.1.SL.TZ0.T_4a.ii:
Write down the coordinates of the local minimum point.
19M.2.SL.TZ1.T_4e:
This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
19M.2.SL.TZ1.T_4b.ii:
State the domain of $P$ .
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
21N.2.SL.TZ0.7a.i:
the boat is taken from $A$ to $P$ , and the bus from $P$ to $B$ .
21N.2.SL.TZ0.7b.ii:
Find the value of $x$ so that $T$ is a minimum.
21N.2.SL.TZ0.7c.i:
Find the new value of $x$ so that the total cost $C$ to travel from $A$ to $B$ via $D$ is a minimum.
21N.2.SL.TZ0.8e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.7a:
Determine the value of $m$ .
21N.2.AHL.TZ0.7b:
Given that $P (|X - m| \leq a) = 0.3$ , determine the value of $a$ .
21N.2.AHL.TZ0.9c:
Find the value of $d$ .
21N.2.AHL.TZ0.9f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9g:
A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.
19M.2.SL.TZ1.T_4b.i:
Find the function, $P\left( x \right)$ , that would define this footpath on the map.
17M.2.SL.TZ2.T_6c.ii:
Find $f(2)$ .
17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .

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