For $a = 0$ and $b = 5\pi$, use the mean value theorem to find all possible values of $c$ for the function $g$.

Sketch the graph of $y = g(x)$ on the interval $[0,{\text{ }}5\pi ]$ and hence illustrate the mean value theorem for the function $g$.

$\frac{{g(5\pi ) - g(0)}}{{5\pi  - 0}} =  - 0.6809 \ldots {\text{ }}\left( { = \cos \sqrt {5\pi } } \right)$ (gradient of chord)     (A1)

$g’(x) = \cos \left( {\sqrt x } \right) - \frac{{\sqrt x \sin \left( {\sqrt x } \right)}}{2}$ (or equivalent)     (M1)(A1)

Note:     Award M1 to candidates who attempt to use the product and chain rules.

attempting to solve $\cos \left( {\sqrt c } \right) - \frac{{\sqrt c \sin \left( {\sqrt c } \right)}}{2} =  - 0.6809 \ldots$ for $c$     (M1)

Notes:     Award M1 to candidates who attempt to solve their $g’(c) =$ gradient of chord.

Do not award M1 to candidates who just attempt to rearrange their equation.

$c = 2.26,{\text{ }}11.1$     A1A1

Note:     Condone candidates working in terms of $x$.

[6 marks]

correct graph: 2 turning points close to the endpoints, endpoints indicated and correct endpoint behaviour     A1

Notes:     Endpoint coordinates are not required. Candidates do not need to indicate axes scales.

correct chord     A1

tangents drawn at their values of $c$ which are approximately parallel to the chord     A1A1

Notes:     Award A1A0A1A0 to candidates who draw a correct graph, do not draw a chord but draw 2 tangents at their values of $c$. Condone the absence of their $c -$ values stated on their sketch. However do not award marks for tangents if no $c -$ values were found in (a).

[4 marks]