Convex and concave curves and points of inflectionEdexcel A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=x^{3}-6x^{2}+2$.

   (a)

   Find $\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}$.

   [1 mark]

   • A$3x^{2}-12x$
   • B$6x-12$
   • C$6x$
   • D$6x^{2}-12x$

   (b)

   For which values of $x$ is the curve concave?

   [1 mark]

   • A$x<2$
   • B$x>2$
   • C$x<0$ or $x>4$
   • D$0<x<4$

   (c)

   Find the coordinates of the point of inflection of $C$, justifying that it is a point of inflection.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=x^{4}$ and $g(x)=x^{3}$. For both functions, the first and second derivatives are equal to $0$ at $x=0$.

   (a)

   Which statement about the origin on $y=f(x)$ is correct?

   [1 mark]

   • AIt is a point of inflection, because $f''(0)=0$.
   • BIt is a maximum, because $f'(0)=0$.
   • CIt is not a stationary point.
   • DIt is a minimum, even though $f''(0)=0$, because $f'(x)$ changes from negative to positive.

   (b)

   Which statement about the origin on $y=g(x)$ is correct?

   [1 mark]

   • AIt is a minimum.
   • BIt is a maximum.
   • CIt is a stationary point of inflection.
   • DIt is not a stationary point.

   (c)

   Explain why $f''(0)=0$ is not enough to show that the origin is a point of inflection on $y=f(x)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve has equation $y=x^{4}-8x^{3}+18x^{2}$.

   (a)

   Find the coordinates of the points of inflection of the curve, justifying your answer.

   [3 marks]

   (b)

   (i) Find the values of $x$ for which the curve is concave.
   (ii) Show that the point on the curve with $x=3$ is a stationary point of inflection.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number of active users, $N$ thousand, of a new app $t$ weeks after its launch is modelled by $N=3t^{2}-\frac13t^{3}$ for $0\leq t\leq6$.

   (a)

   (i) Find $\frac{\mathrm{d}N}{\mathrm{d}t}$ and $\frac{\mathrm{d}^{2}N}{\mathrm{d}t^{2}}$.
   (ii) Show that the graph of $N$ against $t$ has a point of inflection at $t=3$, and find the value of $N$ there.
   (iii) Interpret the significance of $t=3$ for the growth in the number of users.

   [6 marks]

   (b)

   (i) State, with a reason, whether $N$ is convex or concave for $3<t\leq6$, and describe what this means for the growth in users.
   (ii) Find the greatest value of $N$ predicted by the model.
   (iii) A student claims that the model can be used to predict the number of users $10$ weeks after launch. Evaluate this claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions