0606.14.9: Use the first and second derivative tests to Points of inflexion are not included: past paper questions

Real questions on 0606.14.9: use the first and second derivative tests to points of inflexion are not included

The newest 7 of 7 written questions. Open one to see it on the real paper.

1. 0606/23 · May/June 2024 · Q6b(ii)[2]
   (b) The polynomial $p(x)=x^3+a{x}^2+bx+c$, where $a$, $b$ and $c$ are constants, has remainder $-5$ when divided by $x-1$. The curve $y=p(x)$ has stationary points at $x=\frac{4}{3}$ and $x=2$. (ii) Hence use the second derivative test to show that the stationary point at $x=2$ is a minimum.See it on the paper →
2. 0606/11 · May/June 2022 · Q9c[2]
   A circle, centre $O$ and radius $r$ cm, has a sector $OAB$ of fixed area $10{\text{ cm}}^2$. Angle $AOB$ is $\theta$ radians and the perimeter of the sector is $P$ cm. (c) Determine the nature of this stationary value.See it on the paper →
3. 0606/23 · May/June 2022 · Q12[8]
   12 In this question all lengths are in centimetres. [Diagram: A half-cylinder container open at the top. The radius of the semi-circular cross-section is labelled $x$. The length of the cylinder is labelled $y$.] A container is a half-cylinder, open as shown. It has length $y$ and uniform cross-section of radius $x$. T...See it on the paper →
4. 0606/13 · October/November 2021 · Q11c[2]
   11 A curve has equation $y=\frac{(x^2-5{)}^{\frac{1}{3}}}{x+1}$ for $x>-1$. (c) Explain how you could determine the nature of this stationary point. [You are not required to find the nature of this stationary point.]See it on the paper →
5. 0606/12 · May/June 2020 · Q7c[2]
   (c) Determine the nature of this stationary point.See it on the paper →
6. 0606/13 · May/June 2020 · Q10c[2]
   (c) Determine the nature of this stationary point.See it on the paper →
7. 0606/21 · May/June 2020 · Q12b[2]
   (b) A curve has equation $y=f(x)$ and has exactly two stationary points. Given that $f^{\prime \prime }(x)=4x-7$, $f^{\prime }(0.5)=0$ and $f^{\prime }(3)=0$, use the second derivative test to determine the nature of each of the stationary points of this curve.See it on the paper →

What examiners said

Again, candidates who answered parts (a) and (b) well, were generally able to get the method mark here. Fewer candidates achieved the accuracy mark. Common errors were substituting their correct value of r into the second derivative but getting an incorrect value or an incorrectly rounded value.

— Examiner report, May/June 2022, 0606/11 Q9(c) (Cambridge)

The simplest way to be successful in this question was to find an expression for y in terms of x, using the volume given, and then use this to form an expression for the surface area of the container in terms of x only. This expression could then be differentiated in order to progress with the solution. A few neat and fully correct solutions were seen. Some candidates attempted to find an expressi...

— Examiner report, May/June 2022, 0606/23 Q12 (Cambridge)

Most candidates attempted to describe a method using the second derivative. Although most d2 y stated that needed to be found, many then omitted to state that the value of x at the stationary dx 2 point then needed to be substituted in. This was an important and necessary point in the explanation and without it, candidates could not gain any marks. Some candidates attempted to describe a method us...

— Examiner report, October/November 2021, 0606/13 Q11(c) (Cambridge)