1. 1
   The function $f$ is defined by $f(x)=x^{3}-12x+1$, for $x\in\mathbb{R}$.

   (a)

   Find the values of $x$ for which $f'(x)=0$.

   [1 mark]

   • A$x=\pm\sqrt{12}$
   • B$x=\pm4$
   • C$x=2$ only
   • D$x=\pm2$

   (b)

   Find the set of values of $x$ for which $f$ is decreasing.

   [1 mark]

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

   (c)

   Find $f'(3)$ and hence state whether $f$ is increasing or decreasing at $x=3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A function $h$ is defined for all $x\in\mathbb{R}$. The graph of its derivative, $y=h'(x)$, crosses the $x$-axis only at $x=-1$ and touches the $x$-axis only at $x=3$. For $x<-1$, $h'(x)<0$; for $-1<x<3$ and for $x>3$, $h'(x)>0$.

   (a)

   Find the set of values of $x$ for which $h$ is decreasing.

   [1 mark]

   • A$x<-1$
   • B$-1<x<3$
   • C$x>3$
   • D$x<-1$ or $x>3$

   (b)

   Which statement about the graph of $y=h(x)$ at $x=3$ is true?

   [1 mark]

   • A$h$ has a local maximum at $x=3$.
   • B$h(3)=0$.
   • CThe tangent to the graph of $h$ at $x=3$ is horizontal, and $h$ is increasing on both sides of $x=3$.
   • D$h$ changes from increasing to decreasing at $x=3$.

   (c)

   It is given that $h(-1)=-5$. Justify that the equation $h(x)=-6$ has no real solutions.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x)=x^{3}-3x^{2}+kx$, for $x\in\mathbb{R}$, where $k$ is a constant.

   (a)

   In this part, $k=-9$. Find the set of values of $x$ for which $f$ is decreasing.

   [3 marks]

   (b)

   Find the set of values of $k$ for which $f'(x)>0$ for all $x\in\mathbb{R}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The volume of water, $V$ thousand m$^{3}$, in a reservoir $t$ months after the start of a year is modelled by $V(t)=t^{3}-15t^{2}+63t+200$, for $0\le t\le12$.

   (a)

   Find $V'(t)$ and hence find the period of the year during which the volume of water in the reservoir is decreasing.

   [6 marks]

   (b)

   (i) Find $V'(5)$ and interpret your answer in context.
   (ii) The water company claims that the reservoir holds more water at the end of the year than at any other time during the year. Determine whether this claim is correct, justifying your answer.

   [6 marks]

   Total for question 4: 12 marks

End of questions