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Date May 2018 Marks available 4 Reference code 18M.1.AHL.TZ1.H_9
Level Additional Higher Level Paper Paper 1 Time zone Time zone 1
Command term Sketch Question number H_9 Adapted from N/A

Question

Let f(x)=23x52x3,xR,x0.

The graph of y=f(x) has a local maximum at A. Find the coordinates of A.

[5]
a.

Show that there is exactly one point of inflexion, B, on the graph of y=f(x).

[5]
b.i.

The coordinates of B can be expressed in the form B(2a,b×23a) where a, bQ. Find the value of a and the value of b.

[3]
b.ii.

Sketch the graph of y=f(x) showing clearly the position of the points A and B.

[4]
c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt to differentiate      (M1)

f(x)=3x43x     A1

Note: Award M1 for using quotient or product rule award A1 if correct derivative seen even in unsimplified form, for example f(x)=15x4×2x36x2(23x5)(2x3)2.

3x43x=0     M1

x5=1x=1     A1

A(1,52)     A1

[5 marks]

a.

f(x)=0     M1

f(x)=12x53(=0)     A1

Note: Award A1 for correct derivative seen even if not simplified.

x=54(=225)     A1

hence (at most) one point of inflexion      R1

Note: This mark is independent of the two A1 marks above. If they have shown or stated their equation has only one solution this mark can be awarded.

f(x) changes sign at x=54(=225)      R1

so exactly one point of inflexion

[5 marks]

b.i.

x=54=225(a=25)      A1

f(225)=23×222×265=5×265(b=5)     (M1)A1

Note: Award M1 for the substitution of their value for x into f(x).

[3 marks]

b.ii.

A1A1A1A1

A1 for shape for x < 0
A1 for shape for x > 0
A1 for maximum at A
A1 for POI at B.

Note: Only award last two A1s if A and B are placed in the correct quadrants, allowing for follow through.

[4 marks]

c.

Examiners report

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

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection
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Topic 2—Functions
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