1. 1
   Let $f(x)=x^{3}-6x^{2}+9x+1$, for $x\in\mathbb{R}$. The graph of $f$ has stationary points at $x=1$ and $x=3$.

   (a)

   Find the coordinates of the local maximum point of the graph of $f$.

   [1 mark]

   • A$(1,\,5)$
   • B$(3,\,1)$
   • C$(2,\,3)$
   • D$(-1,\,-15)$

   (b)

   Find the set of values of $x$ for which the graph of $f$ is concave-down.

   [1 mark]

   • A$x>2$
   • B$1<x<3$
   • C$x<1$ or $x>3$
   • D$x<2$

   (c)

   Use the second derivative to show that the stationary point at $x=3$ is a local minimum.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $g(x)=x^{4}-4x^{3}+5$, for $x\in\mathbb{R}$. Then $g'(x)=4x^{2}(x-3)$ and $g''(x)=12x(x-2)$.

   (a)

   Which statement describes the stationary point of $g$ at $x=0$?

   [1 mark]

   • AA local maximum
   • BA local minimum
   • CA point of inflexion with zero gradient
   • DIt is not a stationary point

   (b)

   The graph of $g$ has a point of inflexion at $x=2$. Find the gradient of the graph at this point.

   [1 mark]

   • A$0$
   • B$-16$
   • C$-11$
   • D$16$

   (c)

   A student says: \"Since $g''(0)=0$, there must be a point of inflexion at $x=0$.\" By considering the function $y=x^{4}$ at $x=0$, explain why this reasoning is not valid.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A packaging company makes open-topped boxes with a square base of side $x$ cm and height $h$ cm. Each box must hold $32\,000$ cm$^{3}$. The total area of card used, $S$ cm$^{2}$, is the area of the base plus the area of the four sides. Ignore the thickness of the card.

   (a)

   Show that $S=x^{2}+\dfrac{128\,000}{x}$.

   [3 marks]

   (b)

   Find the dimensions of the box that use the least card, and justify that your answer gives a minimum.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A company's weekly profit, in thousands of dollars, from making and selling $x$ hundred units of a product is modelled by $P(x)=-x^{3}+9x^{2}-15x-10$, for $0\le x\le8$.

   (a)

   Find the number of units the company should make and sell each week to maximise its profit, and the maximum weekly profit. Justify that this is the greatest profit for $0\le x\le8$.

   [6 marks]

   (b)

   (i) Find the coordinates of the point of inflexion on the graph of $P$, and show that it is a point of inflexion.
   (ii) Find the gradient of the graph at this point, and interpret this value in context.

   [6 marks]

   Total for question 4: 12 marks

End of questions