7 Fig. 7 shows the curve
$$y = 2 x - x \ln x , \text { where } x > 0 .$$
The curve crosses the \(x\)-axis at A , and has a turning point at B . The point C on the curve has \(x\)-coordinate 1 . Lines CD and BE are drawn parallel to the \(y\)-axis.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a4c3f3b-a298-4b13-b97e-b52f8d9d527b-5_531_1262_671_536}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{figure}
- Find the \(x\)-coordinate of A , giving your answer in terms of e .
- Find the exact coordinates of B .
- Show that the tangents at A and C are perpendicular to each other.
- Using integration by parts, show that
$$\int x \ln x \mathrm {~d} x = \frac { 1 } { 2 } x ^ { 2 } \ln x - \frac { 1 } { 4 } x ^ { 2 } + c$$
Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines CD and BE .
\section*{[Question 8 is printed overleaf.]}