0606.14.6: Use differentiation to find stationary points: past paper questions

Real questions on 0606.14.6: use differentiation to find stationary points

The newest 20 of 56 written questions. Open one to see it on the real paper.

1. 0606/11 · May/June 2026 · Q6a[2]
   The function $f$ is such that $f(x)=3x^2+8x$ for $x⩾k$, where $k$ is a constant. It is given that $f^{-1}$ exists. $k$ has the least possible value for which $f^{-1}$ exists. (a) Find the value of $k.$See it on the paper →
2. 0606/11 · May/June 2026 · Q7b[4]
   7 A curve has equation $y=\frac{4+{\mathrm{e}}^{2x}}{2+{\mathrm{e}}^x}$. (b) Hence find the $x$-coordinate of the stationary point on the curve. Give your answer in the form $\ln (a\sqrt{b}+c)$, where $a$, $b$ and $c$ are integers.See it on the paper →
3. 0606/22 · May/June 2026 · Q9a[5]
   A particle $P$ moves in a straight line such that $t$ seconds after passing through a fixed point $O$, its displacement from $O$, $s$ metres, is given by $s={\mathrm{e}}^{4t}-10{\mathrm{e}}^{2t}+9$. (a) The velocity of $P$ is $v{\mathrm{ms}}^{-1}$. Find the value of $t$ for which $v$ has a stationary value.See it on the paper →
4. 0606/12 · February/March 2026 · Q5[7]
   5 A curve has equation $y=\frac{x^2-8}{3-x}$. $P$ and $Q$ are the stationary points on the curve. Find $\frac{dy}{dx}$ and the exact length of the line $PQ$.See it on the paper →
5. 0606/22 · February/March 2026 · Q7a[5]
   A ship is sailing with speed $v{\text{ kmh}}^{-1}$. The sailing cost per hour, $C$, is given by $C=v^2+\frac{3000}{v}+100$. (a) Find the speed that makes $C$ a minimum. Justify that this value of $C$ is a minimum.See it on the paper →
6. 0606/12 · October/November 2025 · Q9a[11]
   A curve $y=f(x)$ is such that $f^{\prime \prime }(x)=(2x+5{)}^{-\frac{3}{2}}$. The curve has gradient $\frac{2}{3}$ at the point $(2,2)$. (a) Find the coordinates of the stationary point on the curve.See it on the paper →
7. 0606/12 · October/November 2025 · Q9b[1]
   A curve $y=f(x)$ is such that $f^{\prime \prime }(x)=(2x+5{)}^{-\frac{3}{2}}$. The curve has gradient $\frac{2}{3}$ at the point $(2,2)$. (b) Determine the nature of this stationary point.See it on the paper →
8. 0606/21 · October/November 2025 · Q7[6]
   7 Show that the curve $y=x-\ln (x^2+2x)$ has exactly one stationary point. Find the $x$-coordinate of this point.See it on the paper →
9. 0606/21 · October/November 2025 · Q11b[5]
   11 A cylinder has radius $r$ cm and height $h$ cm. The total surface area, including the two ends, is $A$ cm${}^2$. The volume of the cylinder is $330$ cm${}^3$. (b) Given that $r$ can vary, find the value of $r$ that gives a stationary value for $A$ and show that this value is a minimum.See it on the paper →
10. 0606/23 · October/November 2025 · Q10a[6]
    In this question the units are metres and seconds. A particle moves along a straight line through a point $A$. Its displacement, $s$, from $A$ at time $t$ is given by $s=\frac{10t+100}{\sqrt{2t^2+100}}$. The diagram shows the displacement–time graph for the first 30 seconds of the motion. [Diagram: A graph with h...See it on the paper →
11. 0606/23 · October/November 2025 · Q10b(ii)[3]
    10 In this question the units are metres and seconds. A particle moves along a straight line through a point $A$. Its displacement, $s$, from $A$ at time $t$ is given by $s=\frac{10t+100}{\sqrt{2t^2+100}}$. The diagram shows the displacement–time graph for the first 30 seconds of the motion. [Diagram: A graph with ho...See it on the paper →
12. 0606/11 · May/June 2025 · Q6b(ii)[2]
    6 A curve has equation $y={\left(\frac{x^2-1}{x^2+1}\right)}^4$. (ii) Use the first derivative test to determine which two stationary points have the same nature and state whether they are maximum or minimum points.See it on the paper →
13. 0606/12 · May/June 2025 · Q12[6]
    12 A curve is such that its gradient at the point $(x,y)$ is given by $(5x-2{)}^{\frac{1}{3}}$. The curve passes through the point $\left(2,\frac{32}{5}\right)$. Find the coordinates of the stationary point on the curve.See it on the paper →
14. 0606/23 · May/June 2025 · Q12b[5]
    The function f is defined by $f(x)=15{\cos }^2(3x+1.5)+7\sin (3x+1.5)-13$ for $-0.3\le x\le 0.5$, where $x$ is in radians. (b) Find the $x$-coordinates of the two stationary points on the curve $y=f(x)$.See it on the paper →
15. 0606/22 · February/March 2025 · Q9a[4]
    In this question, all lengths are in metres and time is in seconds. A particle, $P$, moves in a straight line such that $t$ seconds after passing through a fixed point $O$ its displacement, $s$, is given by $s=5\ln (2t+1)-5t$. (a) Find the value of $t$ for which $P$ is instantaneously at rest.See it on the paper →
16. 0606/11 · October/November 2024 · Q9b[3]
    The equation of a curve is $y=\frac{{\mathrm{e}}^{-3x+2}}{x+1}$ where $x<-1$. (b) Hence show that there is only one stationary point on the curve and find its exact coordinates.See it on the paper →
17. 0606/13 · October/November 2024 · Q1a[3]
    (a) Find the coordinates of the stationary point on the curve $y=(x+3)(x-4)$.See it on the paper →
18. 0606/22 · October/November 2024 · Q2c[2]
    Variables $x$ and $y$ are related by the equation $y=x\sqrt{1+2x}$. (c) Find the $x$-coordinate of the stationary point on the curve $y=x\sqrt{1+2x}$.See it on the paper →
19. 0606/23 · October/November 2024 · Q4b[5]
    The diagram shows a design for a logo. The logo is a sector of a circle, radius $r$ cm, with angle $\alpha$ radians. [Diagram: A sector of a circle with radius labelled $r$ cm and central angle labelled $\alpha$] The area of the logo is $9{\text{ cm}}^2$. (b) Given that $r$ can vary, find the stationary value of $P$ and...See it on the paper →
20. 0606/12 · May/June 2024 · Q8b[3]

    3-mark “Find” question — its maths and diagrams are best seen on the paper.

    See it on the paper →

What examiners said

Comprehensive responses • Many candidates gave fully correct responses. • Many candidates organised their response well, stating the general form of the quotient rule and then listing the four components they needed. • A few candidates rewrote the function as a product and successfully used the product rule. • Most candidates who had stated a correct derivative went on to form −x2 + 6x − 8 = 0, or...

— Examiner report, February/March 2026, 0606/12 Q5 (Cambridge)

Comprehensive responses • Many fully correct solutions were seen. Limited responses • Some candidates did not show that the value of v gave a minimum point. • Some candidates made errors in differentiation of C with respect to v. • Some candidate rounded their answer to as 11.5 rather than 11.4. • Some candidates did not show enough reasoning when showing the value of C to be a minimum.

— Examiner report, February/March 2026, 0606/22 Q7(a) (Cambridge)

Few candidates attempted to find gradients other than at the three specified points, where the gradient was already known to be zero. Those who assessed the gradient on either side of each point were generally successful, though many candidates would benefit from further practice in explaining their conclusions.

— Examiner report, May/June 2025, 0606/11 Q6(b)(ii) (Cambridge)