1 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.
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\includegraphics[alt={},max width=\textwidth]{74cc215f-bd55-489d-aa4b-0f67c2c8de52-1_529_1259_657_602}
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\caption{Fig. 7}
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- 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 .