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Date	May 2021	Marks available	1	Reference code	21M.3.AHL.TZ1.1
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 1
Command term	Write down	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 5 —Calculus » SL 5.3—Differentiating polynomials, n E Z

Show 91 related questions

18N.2.SL.TZ0.T_6a:
Calculate the area of cloth, in cm², needed to make Haruka’s bag.
19M.2.SL.TZ2.T_5b:
Write down the $y$ -intercept of the graph of $y = f\left( x \right)$ .
SPM.1.SL.TZ0.8b:
Find the value of $a$ and the value of $b$ .
18M.2.SL.TZ1.S_1c:
Solve f '(x) = f "(x).
16N.1.SL.TZ0.S_10c:
(i) Find $h'(x)$ .

(ii) Hence, show that $h'(\pi ) = \frac{{ - 21!}}{2}{\pi ^2}$ .
16N.1.SL.TZ0.S_10b:
(i) Find the first three derivatives of $g(x)$ .

(ii) Given that ${g^{(19)}}(x) = \frac{{k!}}{{(k - p)!}}({x^{k - 19}})$ , find $p$ .
17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation $f(x) = 5$ .
18M.1.AHL.TZ1.H_7a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .
SPM.1.SL.TZ0.8c.i:
Sketch the graph of $y = f'\left( x \right)$ .
17M.2.AHL.TZ1.H_12c:
Explain why $f$ is an even function.
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.
16N.1.SL.TZ0.S_10a:
(i) Find the first four derivatives of $f(x)$ .

(ii) Find ${f^{(19)}}(x)$ .
SPM.1.SL.TZ0.8a:
Find $f'\left( x \right)$ .
19N.2.SL.TZ0.S_3b:
Let ${\text{C}}$ be a point on the graph of $h$ . The tangent to the graph of $h$ at ${\text{C}}$ is parallel to the graph of $f$ .

Find the $x$ -coordinate of ${\text{C}}$ .
19M.2.SL.TZ2.T_5c:
Sketch the graph of $y = f\left( x \right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.
19N.2.SL.TZ0.S_3a:
Find $h\left( x \right)$ .
18N.1.SL.TZ0.S_10b.i:
Find $f'\left( x \right)$ .
16N.1.SL.TZ0.T_14b:
Find the coordinates of P.
18M.1.SL.TZ2.S_9b:
Show that $C = 20\pi {r^2} + \frac{{320\pi }}{r}$ .
19M.1.SL.TZ2.S_10c:
Write down an expression for the area of $R$ .
18N.1.SL.TZ0.S_10c:
The graph of $f$ has a local minimum at the point Q. The line L passes through Q.

Find the value of $a$ .
17M.2.AHL.TZ1.H_2a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$ .
22M.1.SL.TZ2.7d:
Find the values of $d$ for which $g$ is an increasing function.
SPM.1.SL.TZ0.8c.ii:
Hence explain why the graph of $f$ has a local maximum point at $x = a$ .
19M.1.AHL.TZ1.H_7:
Find the coordinates of the points on the curve ${y^3} + 3x{y^2} - {x^3} = 27$ at which $\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$ .
20N.1.SL.TZ0.T_13b:
Write down the gradient of this tangent.
SPM.1.SL.TZ0.8d.i:
Find $f''\left( b \right)$ .
17N.1.SL.TZ0.S_8d:
Find the area of the region enclosed by the graph of $f$ and the line $L$ .
17M.2.AHL.TZ1.H_8b:
Calculate $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ when $\theta = \frac{\pi }{3}$ .
19M.1.SL.TZ2.S_10d:
Hence find the exact area of $R$ .
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 $g'(x) = 0$ ;
17N.1.AHL.TZ0.H_7:
The folium of Descartes is a curve defined by the equation ${x^3} + {y^3} - 3xy = 0$ , shown in the following diagram.

Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the $y$ -axis.
17N.2.SL.TZ0.T_5b.ii:
Find $f’(x)$ .
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
20N.1.SL.TZ0.S_10b:
Find the area of triangle $AOB$ in terms of $k$ .
18M.1.SL.TZ1.S_7:
Consider f(x), g(x) and h(x), for x∈ $\mathbb{R}$ where h(x) = $\left( {f \circ g} \right)$ (x).

Given that g(3) = 7 , g′ (3) = 4 and f ′ (7) = −5 , find the gradient of the normal to the curve of h at x = 3.
17M.2.SL.TZ1.S_6:
Let $f(x) = {({x^2} + 3)^7}$ . Find the term in ${x^5}$ in the expansion of the derivative, $f’(x)$ .
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
18M.1.SL.TZ2.S_9a:
Express h in terms of r.
20N.1.SL.TZ0.S_10c:
The graph of $f$ is translated by $(\begin{matrix} 4 \\ 3 \end{matrix})$ to give the graph of $g$ .
In the following diagram:
- point $Q$ lies on the graph of $g$
- points $C$ , $D$ and $E$ lie on the vertical asymptote of $g$
- points $D$ and $F$ lie on the horizontal asymptote of $g$
- point $G$ lies on the $x$ -axis such that $FG$ is parallel to $DC$ .
Line $L_{2}$ is the tangent to the graph of $g$ at $Q$ , and passes through $E$ and $F$ .

Given that triangle $EDF$ and rectangle $CDFG$ have equal areas, find the gradient of $L_{2}$ in terms of $p$ .
19M.1.SL.TZ2.T_15c:
Hence, find the number of vases that will maximize the profit.
17M.2.AHL.TZ1.H_12e:
Find the inverse function ${g^{ - 1}}$ and state its domain.
20N.1.SL.TZ0.S_10a.ii:
Show that the equation of $L_{1}$ is $k x + p^{2} y - 2 p k = 0$ .
SPM.1.SL.TZ0.8e:
The normal to the graph of $f$ at $x = a$ and the tangent to the graph of $f$ at $x = b$ intersect at the point ( $p$ , $q$ ) .

Find the value of $p$ and the value of $q$ .
20N.1.SL.TZ0.S_10a.i:
Find $f' (p)$ in terms of $k$ and $p$ .
19M.1.SL.TZ1.T_15a:
Write down an equation for the area, $A$ , of the curved surface in terms of $r$ .
19M.1.SL.TZ2.T_15a:
Find the value of $P$ if no vases are sold.
17M.2.SL.TZ2.T_6d.i:
Write down the $x$ -coordinates of these two points;
17N.2.SL.TZ0.T_5b.i:
Expand the expression for $f(x)$ .
17M.1.SL.TZ1.S_9c:
The line $y = kx - 5$ is a tangent to the curve of $f$ . Find the values of $k$ .
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
16N.2.SL.TZ0.T_6c:
Write down, in terms of $r$ and $h$ , an equation for the volume of this water container.
16N.2.SL.TZ0.T_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
16N.2.SL.TZ0.T_6f:
Using your answer to part (e), find the value of $r$ which minimizes $A$ .
17M.2.AHL.TZ1.H_8a:
Find an expression for the volume of water $V{\text{ }}({{\text{m}}^3})$ in the trough in terms of $\theta$ .
18N.1.SL.TZ0.S_10b.ii:
Hence, find the equation of L in terms of $a$ .
18M.1.SL.TZ2.S_9c:
Given that there is a minimum value for C, find this minimum value in terms of $\pi$ .
16N.2.AHL.TZ0.H_6:
An earth satellite moves in a path that can be described by the curve $72.5{x^2} + 71.5{y^2} = 1$ where $x = x(t)$ and $y = y(t)$ are in thousands of kilometres and $t$ is time in seconds.

Given that $\frac{{{\text{d}}x}}{{{\text{d}}t}} = 7.75 \times {10^{ - 5}}$ when $x = 3.2 \times {10^{ - 3}}$ , find the possible values of $\frac{{{\text{d}}y}}{{{\text{d}}t}}$ .

Give your answers in standard form.
SPM.1.SL.TZ0.8d.ii:
Hence, use your answer to part (d)(i) to show that the graph of $f$ has a local minimum point at $x = b$ .
17M.2.AHL.TZ1.H_2b:
Determine the equation of the tangent to $C$ at the point $\left( {\frac{2}{{\text{e}}},{\text{ e}}} \right)$
19M.1.SL.TZ2.S_10b:
Hence find $\int {\left( {3{x^2} + 1} \right)\sqrt {{x^3} + x} } \,{\text{d}}x$ .
18M.1.AHL.TZ1.H_7b:
Find $\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x}$ .
17M.2.AHL.TZ1.H_12g.ii:
Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .
16N.1.SL.TZ0.T_14a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .
18M.1.AHL.TZ2.H_6b:
Hence, or otherwise, find $\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x}$ .
18N.2.SL.TZ0.T_6g:
Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.
19M.1.SL.TZ1.T_15c:
Find the value of $r$ when the area of the curved surface is maximized.
18N.2.SL.TZ0.T_6d:
Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.
19M.1.SL.TZ2.T_15b:
Differentiate $P\left( x \right)$ .
20N.1.SL.TZ0.T_13c:
Find the value of $k$ .
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
20N.1.SL.TZ0.T_13a:
Write down $f' (x)$ .
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.
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
16N.2.SL.TZ0.T_6d:
Show that $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ .
16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
19M.1.SL.TZ1.T_15b:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
18N.2.SL.TZ0.T_6e:
Use your answers to parts (c) and (d) to show that

$A = 3{x^2} + \frac{{10368}}{x}$ .
16N.2.SL.TZ0.T_6a:
Write down a formula for $A$ , the surface area to be coated.
16N.2.SL.TZ0.T_6h:
Find the least number of cans of water-resistant material that will coat the area in part (g).
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.
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 $f(x)$ .
18N.2.SL.TZ0.T_6b:
Calculate the volume, in cm³, of the bag.
18N.2.SL.TZ0.T_6c:
Use this value to write down, and simplify, the equation in x and y for the volume of Nanako’s bag.
18N.2.SL.TZ0.T_6f:
Find $\frac{{{\text{d}}A}}{{{\text{d}}x}}$ .

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