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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 x3-3cx+d.

 

Consider the function fx=x3-3cx+2 for x and where c is a parameter, c.

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]
b.

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 2c32+2.

[3]
d.i.

the y-coordinate of the local minimum point is -2c32+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)=x3-3cx+d for x and where c , d.

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]
f.

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 -2,2+42.
          Local minimum point coordinates 2,2-42.

 

[3 marks]

a.ii.

f'(x) =3x2-3c       A1

 

Note: Accept 3x2-3c (an expression).

 

[1 mark]

b.

c=0       A1

 

[1 mark]

c.i.

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

3x2-3c=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±c        (A1)

 

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

 

correctly evaluates f-c        A1  

f-c=-c32+3c32+2 =-cc+3cc+2

the y-coordinate of the local maximum point is 2c32+2        AG

 

[3 marks]

d.i.

 

correctly evaluates fc        A1  

fc=c32-3c32+2 =cc-3cc+2

the y-coordinate of the local minimum point is -2c32+2        AG

 

[1 mark]

d.ii.

the graph of y=fx will have one x-axis intercept if

EITHER

-2c32+2>0 (or equivalent reasoning)         R1

 

OR

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=fx will have two x-axis intercepts if

EITHER

-2c32+2=0 (or equivalent reasoning)         (M1)

 

OR

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

 

THEN

c=1        A1  

  

[2 marks]

e.ii.

the graph of y=fx will have three x-axis intercepts if

EITHER

-2c32+2<0 (or equivalent reasoning)         (M1)

 

OR

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

 

THEN

c>1        A1  

  

[2 marks]

e.iii.

case 1:

c0 (independent of the value of d)        A1 

EITHER

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


OR

g'x0  for  x~                   R1


OR

the graph of y=fx has no local maximum or local minimum points, hence any vertical translation of this graph (y=gx) 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

-c,2c32+d is a local maximum point and c,-2c32+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>2c32 (both points above the x-axis)        A1 

d<-2c32 (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]

f.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.
[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]
f.

Syllabus sections

Topic 2—Functions » SL 2.3—Graphing
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