(a) Sketch of \( y = \left| \ln x \right| – \left| \cos x \right| – 0.1 \):

Correct shape, with \( \left| \ln x \right| \) increasing logarithmically and \( \left| \cos x \right| \) oscillating, shifted by \(-0.1\): \( (A1) \)

x-intercepts: Solve \( \left| \ln x \right| = \left| \cos x \right| + 0.1 \) numerically or graphically:

\[ \begin{cases} 0.354 \\ 1.36 \\ 2.59 \\ 2.95 \end{cases} \quad (A2) \]

Note: Award A1 for three correct x-intercepts, A0 otherwise.

Maximum: Compute the derivative (considering cases due to absolute values) to find the critical point at \( x = \frac{\pi}{2} \approx 1.57 \), evaluate \( y \approx 0.352 \):

\[ \left( \frac{\pi}{2}, 0.352 \right) \quad (A1) \]

Minima: Identify critical points or use graphical analysis to find minima:

\[ \begin{cases} (1, -0.640) \\ (2.77, -0.0129) \end{cases} \quad (A1) \]

[5 marks]

(b) Values of \( x \) for which \( \left| \ln x \right| > \left| \cos x \right| + 0.1 \):

Find where \( y = \left| \ln x \right| – \left| \cos x \right| – 0.1 > 0 \), using x-intercepts from part (a) to determine intervals:

\[ \begin{cases} 0 < x < 0.354 \\ 1.36 < x < 2.59 \\ 2.95 < x < 4 \end{cases} \quad (A2) \]

Note: Award A1 if two correct regions are given.

[2 marks]