6.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{79c1c4df-3812-4295-b01d-4724eda3457d-4_656_791_315_386}
\end{figure}
Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where
$$\mathrm { f } ( x ) = 10 + \ln ( 3 x ) - \frac { 1 } { 2 } \mathrm { e } ^ { x } , 0.1 \leq x \leq 3.3 .$$
Given that \(\mathrm { f } ( k ) = 0\),
- show, by calculation, that \(3.1 < k < 3.2\).
- Find \(\mathrm { f } ^ { \prime } ( x )\).
The tangent to the graph at \(x = 1\) intersects the \(y\)-axis at the point \(P\).
- Find an equation of this tangent.
- Find the exact \(y\)-coordinate of \(P\), giving your answer in the form \(a + \ln b\).