Mathematical Methods · Past QCAA questions All simple familiar questions
2025 · Paper 1 · Simple familiar

Question 13

[4 marks] Technology-free
Topic guidePremiumUnit 3 Topic 3.1 · Concavity and points of inflection
Game planPremium1.8 Point of inflection
The question

Use $f''(x)=2x-6$ to demonstrate that the function $y=f(x)$ has a point of inflection at $x=3$.

Watch the situation first

The second derivative changes sign, so the curve changes bend

The bottom graph is the $f''$ you are given. The top curve is one possible $f$ with that second derivative. Watch how the sign below matches the bend above.

You are given f′′, not f Substitute x = 3 into f′′ Left of 3, f′′ is negative: concave down Right of 3, f′′ is positive: concave up The concavity changes, so x = 3 is a point of inflection one possible y = f(x) 12345 x y = f′′(x) = 2x − 6 concave down concave up f′′(2) = −2 < 0 f′′(3) = 2(3) − 6 = 0 f′′(4) = 2 > 0 point of inflection

$f''(3)=0$ only makes $x=3$ a candidate. It is the change of sign either side that makes it a point of inflection.

Now for the mathematics

Is a zero second derivative enough?

All three second derivatives below are zero at $x=3$. The first is the one in the question. Choose one, move the test points either side, and see whether the sign changes.

one possible y = f(x) y = f′′(x) a b 12345
$x$$f''(x)$Concavity

$f''$ goes from negative to positive at $x=3$, so the curve goes from concave down to concave up. That makes $x=3$ a point of inflection. Every test point gives the same signs, because $2x-6$ is linear and crosses zero only once.

$f''(3)=0$ here too, but $f''$ is positive on both sides. The curve is concave up the whole way through, so $x=3$ is not a point of inflection.

This $f''$ isn’t linear, but it still changes sign at $x=3$, from negative to positive. So $x=3$ is a point of inflection.

QCAA marking guide · 4 marks

Work through the solution one mark at a time

Try each step yourself before you reveal it.

Step 11 mark

At a point of inflection, the second derivative is zero. Substitute $x=3$.

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

Solving $f''(x)=0$ to get $x=3$ is also accepted.

Marker: correctly demonstrates that the value of the second derivative is equal to zero at $x=3$

Step 21 mark

Zero on its own isn’t enough. The second derivative has to change sign about the point. $f''(x)$ is linear, so it changes sign only once. Test a value to the left of $x=3$.

$$f''(2)=2\times2-6$$
$$=-2\lt0$$
$$\text{concave down}$$

Other $x$-values on the left are accepted, such as $f''(1)=-4$.

Marker: correctly establishes that $f''(x)$ is negative to the left of the point

Step 31 mark

Now test a value to the right.

$$f''(4)=2\times4-6$$
$$=2\gt0$$
$$\text{concave up}$$

Other $x$-values on the right are accepted, such as $f''(3.5)=1$.

Marker: correctly establishes that $f''(x)$ is positive to the right of the point

Step 41 mark

Now state the conclusion, and say what it rests on.

As the sign of the second derivative changes around $x=3$, the function has a point of inflection at $x=3$.

This mark can only be awarded if your earlier working supports the decision. A conclusion with no sign test behind it doesn’t get the mark.

Marker: correctly concludes the point $x=3$ is an inflection point

Putting it all together

Each phrase of the question gave you something

“Use $f''(x)=2x-6$”
You work only with the second derivative. You never need $f$ itself.
“to demonstrate”
The answer is given, so the marks are for the working: the value at $x=3$, a test on each side, and a conclusion.
“has a point of inflection”
The concavity must change. That means $f''=0$ and $f''$ changes sign.
“at $x=3$”
Substitute $x=3$ itself, then one value either side of it.
What makes this simple familiar

The second derivative is given, and it is linear. The whole question is three substitutions and a sentence.

If I had to guess where you might lose marks, it would be stopping after $f''(3)=0$. That only makes $x=3$ a candidate. $f''(x)=(x-3)^2$ is also zero at $x=3$, but it never changes sign, so there is no point of inflection there. Show the sign change on both sides, then write the conclusion.

Only mark this done when you could do it without help. Reading the solution does not count.

Back to the index

Question wording and marking-guide steps are from the 2025 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.

Spot a mistake on this page? Report it