本题目来源于试卷: Proofs Proofs ，类别为 IB数学
[问答题]
Let $f(x)=(x-1) e^{\frac{x}{3}}$ , for $x \in \mathbb{R}$ .
1. Find $f^{\prime}(x)$ .
2. Prove by induction that $\frac{\mathrm{d}^{n} f}{\mathrm{~d} x^{n}}=\left(\frac{3 n+x-1}{3^{n}}\right) e^{\frac{x}{3}}$ for all $n \in \mathbb{Z}^{+}$ .
3. Find the coordinates of any local maximum and minimum points on the graph of y=f(x) . Justify whether such point is a maximum or a minimum.
4. Find the coordinates of any points of inflexion on the graph of y=f(x) . Justify whether such point is a point of inflexion.
5. Hence sketch the graph of $y=f(x)$ , indicating clearly the points found in parts (c) and (d) and any intercepts with the axes.
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