本题目来源于试卷: Differential Calculus，类别为 IB数学
[问答题]
Consider the function ljm.+ax odj04 y3v5ua:k) egmf(/ka x+kf 5s; wu7i3whu 2job $f(x)=\frac{a e^{-x}}{b-a e^{-x}}$ where $\lt 0, b\lt0$ .
1. Show that $ f^{\prime}(x)=\frac{-a b e^{-x}}{\left(b-a e^{-x}\right)^{2}} $.
2. Explain why $f^{\prime \prime}(x)$ is never zero.
3. Find the equation of:
1. the vertical asymptote of f ;
2. the horizontal asymptote of f .
4. Draw a sign diagram for $f^{\prime}(x)$ .
5. If a=3 and b=1 ,
1. sketch the graph of f labelling all asymptotes;
2. find the area of the region enclosed by f , the x and y axes and the line $x=\ln 2 $.
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