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Date May 2019 Marks available 3 Reference code 19M.2.SL.TZ1.S_4
Level Standard Level Paper Paper 2 Time zone Time zone 1
Command term Sketch Question number S_4 Adapted from N/A

Question

Let f(x)=(cos2x)(sin6x), for 0 ≤ x ≤ 1.

Sketch the graph of f on the grid below:

[3]
a.

Find the x-coordinates of the points of inflexion of the graph of f.

[3]
b.

Hence find the values of x for which the graph of f is concave-down.

[2]
c.

Markscheme

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

  A1A1A1 N3

Note: Only if the shape is approximately correct with exactly 2 maximums and 1 minimum on the interval 0 ≤ x ≤ 0, award the following:
A1 for correct domain with both endpoints within circle and oval.
A1 for passing through the other x-intercepts within the circles.
A1 for passing through the three turning points within circles (ignore x-intercepts and extrema outside of the domain).

[3 marks]

a.

evidence of reasoning (may be seen on graph)      (M1)

eg  f=0,  (0.524, 0),  (0.785, 0)

0.523598,  0.785398

x=0.524(=π6),  x=0.785(=π4)     A1A1  N3

Note: Award M1A1A0 if any solution outside domain (eg x=0) is also included.

[3 marks]

b.

0.524<x<0.785(π6<x<π4)     A2  N2

Note: Award A1 if any correct interval outside domain also included, unless additional solutions already penalized in (b).
Award A0 if any incorrect intervals are also included.

[2 marks]

c.

Examiners report

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

Syllabus sections

Topic 5 —Calculus » SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules
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Topic 5 —Calculus » SL 5.8—Testing for max and min, optimisation. Points of inflexion
Topic 5 —Calculus

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