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]