Find the exact value of each of the zeros of $f$.

Expand the expression for $f(x)$.

Find $f’(x)$.

Use your answer to part (b)(ii) to find the values of $x$ for which $f$ is increasing.

Draw the graph of $f$ for $- 3 \leqslant x \leqslant 3$ and $- 40 \leqslant y \leqslant 20$. Use a scale of 2 cm to represent 1 unit on the $x$-axis and 1 cm to represent 5 units on the $y$-axis.

Write down the coordinates of the point of intersection.

$- 1,{\text{ }}\sqrt 5 ,{\text{ }} - \sqrt 5$     (A1)(A1)(A1)

Note:     Award (A1) for –1 and each exact value seen. Award at most (A1)(A0)(A1) for use of 2.23606… instead of $\sqrt 5$.

[3 marks]

$10x - 2{x^3} + 10 - 2{x^2}$     (A1)

Notes:     The expansion may be seen in part (b)(ii).

[1 mark]

$10 - 6{x^2} - 4x$     (A1)(ft)(A1)(ft)(A1)(ft)

Notes:     Follow through from part (b)(i). Award (A1)(ft) for each correct term. Award at most (A1)(ft)(A1)(ft)(A0) if extra terms are seen.

[3 marks]

$10 - 6{x^2} - 4x > 0$     (M1)

Notes:     Award (M1) for their $f’(x) > 0$. Accept equality or weak inequality.

$- 1.67 < x < 1{\text{ }}\left( { - \frac{5}{3} < x < 1,{\text{ }} - 1.66666 \ldots  < x < 1} \right)$     (A1)(ft)(A1)(ft)(G2)

Notes:     Award (A1)(ft) for correct endpoints, (A1)(ft) for correct weak or strict inequalities. Follow through from part (b)(ii). Do not award any marks if there is no answer in part (b)(ii).

[3 marks]

     (A1)(A1)(ft)(A1)(ft)(A1)

Notes:     Award (A1) for correct scale; axes labelled and drawn with a ruler.

Award (A1)(ft) for their correct $x$-intercepts in approximately correct location.

Award (A1) for correct minimum and maximum points in approximately correct location.

Award (A1) for a smooth continuous curve with approximate correct shape. The curve should be in the given domain.

Follow through from part (a) for the $x$-intercepts.

[4 marks]

$(1.49,{\text{ }}13.9){\text{ }}\left( {(1.48702 \ldots ,{\text{ }}13.8714 \ldots )} \right)$     (G1)(ft)(G1)(ft)

Notes:     Award (G1) for 1.49 and (G1) for 13.9 written as a coordinate pair. Award at most (G0)(G1) if parentheses are missing. Accept $x = 1.49$ and $y = 13.9$. Follow through from part (b)(i).

[2 marks]