The function \(f\) is defined by \(f(x) = (\ln x)^2\) for \(x > 0\). The diagram shows a sketch of the graph of \(y = f(x)\). The minimum point of the graph is \(A\). The point \(B\) has \(x\)-coordinate \(e\).

(i) State the \(x\)-coordinate of \(A\).

(ii) Show that \(f''(x) = 0\) at \(B\).

(iii) Use the substitution \(x = e^u\) to show that the area of the region bounded by the \(x\)-axis, the line \(x = e\), and the part of the curve between \(A\) and \(B\) is given by \(\int_0^1 u^2 e^u \, du\).

(iv) Hence, or otherwise, find the exact value of this area.