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]