本题目来源于试卷: Differential Calculus，类别为 IB数学
[问答题]
Let $f(x)=\frac{x^{3}+x-2}{2 x}, x \in \mathbb{R}, x \neq 0$ .
1. The graph of y=f(x) has a local minimum at A . Find the coordinates of A .
2. 1. Show that there is exactly one point of inflexion, B , on the graph of y=f(x) .
2. The coordinates of B can be expressed in the form $\mathrm{B}\left(2^{p}, 2^{q}\right) $, where p, q $\in \mathbb{Q} $. Find the value of p and the value of q .
3. Sketch the graph of y=f(x) showing clearly the position of the points A and B .
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