Turning points of a cubic with a sine term and the area between the curve and a chord through the origin

The function \(\mathrm{g}\) is given by \(\mathrm{g}(x)=x^3-2x+\sin x\) for \(x\in\mathbb{R}\).

The curve \(y=\mathrm{g}(x)\) has a local maximum point \(A\) and a local minimum point \(B\), and the straight line through \(A\) and \(B\) passes through the origin \(O\).

The shaded region \(R\) in the diagram is bounded by the curve and the line segment \([OB]\).

1. Find the coordinates of \(A\) and of \(B\).[2]
2. Find the area of \(R\).[5]