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