Mathematical Methods — Question 16

QCAA 2020, Paper 1 · 4 marks

Q16 · 2020 · Technology-freeSimple familiar4 marks

QUESTION 16 (4 marks)

Consider the graph of

f(x)

shown. Identify the graph of the second derivative

f''(x)

from the graphs in Diagram 1, Diagram 2 and Diagram 3. Justify your decisions using mathematical reasoning.

WORKED SOLUTION

4 marks

ANSWER

Diagram 1

Worked solution

f(x)

changes concavity at certain points and is increasing and decreasing between

x=0

and

x=3

, and between

x=3

and

x=6

. Therefore the sketch of

f'(x)

must be positive and negative over these intervals.

f'(x)

has maximum and minimum points at approximately

x=0.8

and

x=2.4

, so

f''(x)

crosses the

x

-axis at these points. Diagram 1 is

f''(x)

correctly identifies an appropriate method to determine the

f''(x)

graph

[1 mark]

correctly determines a relevant feature of the graph of

f(x)

[1 mark]

correctly determines a relevant feature for

f''(x)

[1 mark]

correctly identifies Diagram 1

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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