Mathematical Methods — Question 13

QCAA 2025, Paper 1 · 4 marks

Q13 · 2025 · Technology-freeSimple familiar4 marks

QUESTION 13 (4 marks)

Use

f''(x)=2x-6

to demonstrate that the function

y=f(x)

has a point of inflection at

x=3

WORKED SOLUTION

4 marks

ANSWER

x=3

is a point of inflection.

Worked solution

Confirm that

f''(x)=0

x=3

f''(3)=2\times3-6=0.

To verify a point of inflection, there must be a change of sign of the second derivative. Since

f''(x)

is linear, test values either side of

x=3

f''(2)=-2<0\quad\text{(concave down)},

f''(4)=2>0\quad\text{(concave up)}.

As the sign of the second derivative changes around

x=3

, the function has a point of inflection there.

correctly demonstrates that the value of the second derivative is equal to zero at

x=3

[1 mark]

correctly establishes that

f''(x)

is negative to the left of the point

[1 mark]

correctly establishes that

f''(x)

is positive to the right of the point

[1 mark]

correctly concludes the point

x=3

is an inflection point

[1 mark]

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

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