(i)     Solve the equation $f'(x) = 0$ .

(ii)     Hence show the graph of $f$ has a local maximum.

(iii)     Write down the range of the function $f$ .

Show that there is a point of inflexion on the graph and determine its coordinates.

Sketch the graph of $y = f(x)$ , indicating clearly the asymptote, x-intercept and the local maximum.

Now consider the functions $g(x) = \frac{{\ln \left| x \right|}}{x}$ and $h(x) = \frac{{\ln \left| x \right|}}{{\left| x \right|}}$ , where $0 < x < {{\text{e}}^2}$ .

(i)     Sketch the graph of $y = g(x)$ .

(ii)     Write down the range of $g$ .

(iii)     Find the values of $x$ such that $h(x) > g(x)$ .

(i)     $f'(x) = \frac{{x\frac{1}{x} - \ln x}}{{{x^2}}}$     M1A1

$= \frac{{1 - \ln x}}{{{x^2}}}$

so $f'(x) = 0$ when $\ln x = 1$, i.e. $x = {\text{e}}$     A1

(ii)     $f'(x) > 0$ when $x < {\text{e}}$ and $f'(x) < 0$ when $x > {\text{e}}$     R1

hence local maximum     AG

Note: Accept argument using correct second derivative.

(iii)     $y \leqslant \frac{1}{{\text{e}}}$     A1

[5 marks]

$f''(x) = \frac{{{x^2}\frac{{ - 1}}{x} - \left( {1 - \ln x} \right)2x}}{{{x^4}}}$     M1

$= \frac{{ - x - 2x + 2x\ln x}}{{{x^4}}}$

$= \frac{{ - 3 + 2\ln x}}{{{x^3}}}$     A1

Note: May be seen in part (a).

$f''(x) = 0$     (M1)

${ - 3 + 2\ln x = 0}$

$x = {{\text{e}}^{\frac{3}{2}}}$

since $f''(x) < 0$ when $x < {{\text{e}}^{\frac{3}{2}}}$ and $f''(x) > 0$ when $x > {{\text{e}}^{\frac{3}{2}}}$     R1

then point of inflexion $\left( {{{\text{e}}^{\frac{3}{2}}},\frac{3}{{2{{\text{e}}^{\frac{3}{2}}}}}} \right)$     A1

[5 marks]

     A1A1A1

Note: Award A1 for the maximum and intercept, A1 for a vertical asymptote and A1 for shape (including turning concave up).

[3 marks]

(i)
     A1A1

Note: Award A1 for each correct branch.

(ii) all real values     A1

(iii)
     (M1)(A1)

Note: Award (M1)(A1) for sketching the graph of h, ignoring any graph of g.

$- {{\text{e}}^2} < x <  - 1$ (accept $x < - 1$ )     A1

[6 marks]

Most candidates attempted parts (a), (b) and (c) and scored well, although many did not gain the reasoning marks for the justification of the existence of local maximum and inflexion point. The graph sketching was poorly done. A wide selection of range shapes were seen, in some cases showing little understanding of the relation between the derivatives of the function and its graph and difficulties with transformation of graphs. In some cases candidates sketched graphs consistent with their previous calculations but failed to label them properly.

Most candidates attempted parts (a), (b) and (c) and scored well, although many did not gain the reasoning marks for the justification of the existence of local maximum and inflexion point. The graph sketching was poorly done. A wide selection of range shapes were seen, in some cases showing little understanding of the relation between the derivatives of the function and its graph and difficulties with transformation of graphs. In some cases candidates sketched graphs consistent with their previous calculations but failed to label them properly.

Most candidates attempted parts (a), (b) and (c) and scored well, although many did not gain the reasoning marks for the justification of the existence of local maximum and inflexion point. The graph sketching was poorly done. A wide selection of range shapes were seen, in some cases showing little understanding of the relation between the derivatives of the function and its graph and difficulties with transformation of graphs. In some cases candidates sketched graphs consistent with their previous calculations but failed to label them properly.

Most candidates attempted parts (a), (b) and (c) and scored well, although many did not gain the reasoning marks for the justification of the existence of local maximum and inflexion point. The graph sketching was poorly done. A wide selection of range shapes were seen, in some cases showing little understanding of the relation between the derivatives of the function and its graph and difficulties with transformation of graphs. In some cases candidates sketched graphs consistent with their previous calculations but failed to label them properly.