1. 1
   Let $f(x)=x^{4}-2x^{3}+5x$, for $x\in\mathbb{R}$.

   (a)

   Find $f''(x)$.

   [1 mark]

   • A$4x^{3}-6x^{2}+5$
   • B$12x^{2}-12x$
   • C$12x^{2}-12x+5$
   • D$12x^{2}-6x$

   (b)

   Find the value of $f''(-1)$.

   [1 mark]

   • A$0$
   • B$-5$
   • C$24$
   • D$-24$

   (c)

   Determine whether the gradient of the graph of $f$ is increasing or decreasing at $x=\frac{1}{2}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve has equation $y=e^{3x}-6x$, for $x\in\mathbb{R}$.

   (a)

   Find $\dfrac{d^{2}y}{dx^{2}}$.

   [1 mark]

   • A$3e^{3x}$
   • B$e^{3x}$
   • C$9e^{3x}-6$
   • D$9e^{3x}$

   (b)

   Which statement about the gradient of the curve is true for all $x$?

   [1 mark]

   • AThe gradient is always increasing.
   • BThe gradient is always positive.
   • CThe gradient is always decreasing.
   • DThe gradient is constant.

   (c)

   Show that $\dfrac{d^{2}y}{dx^{2}}=3\dfrac{dy}{dx}+18$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A function $f$ is defined for all $x\in\mathbb{R}$. Its derivative is $f'(x)=x^{3}-3x^{2}+4$, which can be written as $f'(x)=(x+1)(x-2)^{2}$.

   (a)

   Find $f''(x)$ and hence find the values of $x$ for which $f'$ is increasing.

   [3 marks]

   (b)

   A student notes that $f'(2)=0$ and $f''(2)=0$, and claims that the graph of $f$ has a local maximum at $x=2$. Explain why the student is wrong, and describe in words how the gradient of $f$ behaves as $x$ increases through $2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number of fish in a lake is modelled by $P(t)=400+60t^{2}-2t^{3}$, where $t$ is the time in months after the lake was restocked, $0\le t\le20$.

   (a)

   (i) Find $P'(t)$ and $P''(t)$.
   (ii) Find the time at which the fish population is growing fastest, justifying your answer, and find the rate of growth at this time.

   [6 marks]

   (b)

   (i) Find $P''(5)$ and $P''(15)$, and interpret each value in context.
   (ii) An ecologist says: \"After $t=10$ the number of fish in the lake is falling.\" Evaluate this claim for $10<t<20$.

   [6 marks]

   Total for question 4: 12 marks

End of questions