Obtain an expression for \(f'(x)\) .

Sketch the graphs of f and \(f'\) on the same axes, showing clearly all x-intercepts.

Find the x-coordinates of the two points of inflexion on the graph of f .

Find the equation of the normal to the graph of f where x = 0.75 , giving your answer in the form y = mx + c .

Consider the points \({\text{A}}\left( {a{\text{ }},{\text{ }}f(a)} \right)\) , \({\text{B}}\left( {b{\text{ }},{\text{ }}f(b)} \right)\) and \({\text{C}}\left( {c{\text{ }},{\text{ }}f(c)} \right)\) where a , b and c \((a < b < c)\) are the solutions of the equation \(f(x) = f'(x)\) . Find the area of the triangle ABC.

\(f'(x) = \frac{1}{{x + 1}}\sin (\pi x) + \pi \ln (x + 1)\cos (\pi x)\)     M1A1A1

[3 marks]

     A4

Note: Award A1A1 for graphs, A1A1 for intercepts.

[4 marks]

0.310, 1.12     A1A1

[2 marks]

\(f'(0.75) = - 0.839092\)     A1

so equation of normal is \(y - 0.39570812 = \frac{1}{{0.839092}}(x - 0.75)\)     M1

\(y = 1.19x - 0.498\)     A1

[3 marks]

\({\text{A}}(0,{\text{ }}0)\)

\({\text{B(}}\overbrace {0.548 \ldots }^c,\overbrace {0.432 \ldots }^d)\)     A1

\({\text{C(}}\overbrace {1.44 \ldots }^e,\overbrace { - 0.881 \ldots }^f)\)     A1

Note: Accept coordinates for B and C rounded to 3 significant figures.

area \(\Delta {\text{ABC}} = \frac{1}{2}|\)(ci + dj) \( \times \) (ei + fj)\(|\)     M1A1

\( = \frac{1}{2}(de - cf)\)     A1

\( = 0.554\)     A1

[6 marks]