Write down the $y$-intercept of the graph.

Find $f'(x)$.

Show that $a = 8$.

Find $f(2)$.

Write down the $x$-coordinates of these two points;

Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.

Write down the range of $f(x)$.

Write down the number of possible solutions to the equation $f(x) = 5$.

The equation $f(x) = m$, where $m \in \mathbb{R}$, has four solutions. Find the possible values of $m$.

5     (A1)

Note:     Accept an answer of $(0,{\text{ }}5)$.

[1 mark]

$\left( {f'(x) = } \right) - 4{x^3} + 2ax$     (A1)(A1)

Note:     Award (A1) for $- 4{x^3}$ and (A1) for $+ 2ax$. Award at most (A1)(A0) if extra terms are seen.

[2 marks]

$- 4 \times {2^3} + 2a \times 2 = 0$     (M1)(M1)

Note:     Award (M1) for substitution of $x = 2$ into their derivative, (M1) for equating their derivative, written in terms of $a$, to 0 leading to a correct answer (note, the 8 does not need to be seen).

$a = 8$     (AG)

[2 marks]

$\left( {f(2) = } \right) - {2^4} + 8 \times {2^2} + 5$     (M1)

Note:     Award (M1) for correct substitution of $x = 2$ and  $a = 8$ into the formula of the function.

21     (A1)(G2)

[2 marks]

$(x = ){\text{ }} - 2,{\text{ }}(x = ){\text{ 0}}$     (A1)(A1)

Note:     Award (A1) for each correct solution. Award at most (A0)(A1)(ft) if answers are given as $( - 2{\text{ }},21)$ and $(0,{\text{ }}5)$ or $( - 2,{\text{ }}0)$ and $(0,{\text{ }}0)$.

[2 marks]

$x <  - 2,{\text{ }}0 < x < 2$     (A1)(ft)(A1)(ft)

Note:     Award (A1)(ft) for $x <  - 2$, follow through from part (d)(i) provided their value is negative.

Award (A1)(ft) for $0 < x < 2$, follow through only from their 0 from part (d)(i); 2 must be the upper limit.

Accept interval notation.

[2 marks]

$y \leqslant 21$     (A1)(ft)(A1)

Notes:     Award (A1)(ft) for 21 seen in an interval or an inequality, (A1) for “$y \leqslant$”.

Accept interval notation.

Accept $- \infty  < y \leqslant 21$ or $f(x) \leqslant 21$.

Follow through from their answer to part (c)(ii). Award at most (A1)(ft)(A0) if $x$ is seen instead of $y$. Do not award the second (A1) if a (finite) lower limit is seen.

[2 marks]

3 (solutions)     (A1)

[1 mark]

$5 < m < 21$ or equivalent     (A1)(ft)(A1)

Note:     Award (A1)(ft) for 5 and 21 seen in an interval or an inequality, (A1) for correct strict inequalities. Follow through from their answers to parts (a) and (c)(ii).

Accept interval notation.

[2 marks]