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Date May 2022 Marks available 6 Reference code 22M.1.AHL.TZ2.6
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 2
Command term Find Question number 6 Adapted from N/A

Question

A function f is defined by fx=x1-x2 where -1x1.

The graph of y=f(x) is shown below.

Show that f is an odd function.

[2]
a.

The range of f is ayb, where a, b.

Find the value of a and the value of b.

[6]
b.

Markscheme

attempts to replace x with -x        M1

f-x=-x1--x2

=-x1--x2=-fx         A1

 

Note: Award M1A1 for an attempt to calculate both f-x and -f-x independently, showing that they are equal.
Note: Award M1A0 for a graphical approach including evidence that either the graph is invariant after rotation by 180° about the origin or the graph is invariant after a reflection in the y-axis and then in the x-axis (or vice versa).

 

so f is an odd function         AG

  

[2 marks]

a.

attempts both product rule and chain rule differentiation to find f'x        M1

f'x=x×12×-2x×1-x2-12+1-x212×1 =1-x2-x21-x2         A1

=1-2x21-x2

sets their f'x=0        M1

x=±12         A1

attempts to find at least one of f±12         (M1)

 

Note: Award M1 for an attempt to evaluate fx at least at one of their f'x=0  roots.

 

a=-12  and b=12         A1

 

Note: Award A1 for -12y12.

  

[6 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection
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Topic 5 —Calculus » SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules
Topic 5 —Calculus » SL 5.8—Testing for max and min, optimisation. Points of inflexion
Topic 2—Functions
Topic 5 —Calculus

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