[问答题]
Consider the function
( uw*h(v*gv ov pz5)mr r t ,6mc76 .ztc3 affp.-mlarp $g(x)=a x^{3}+b x^{2}+c x+d$ , where $ x \in \mathbb{R}$ and a, b, c, d $\in \mathbb{R}$ .
1. 1. Write down an expression for $g^{\prime}(x)$ .
2. Hence, given that $ g^{-1}$ does not exist, show that $b^{2}-3$ a c>0 .
Consider the function $f(x)=\frac{x^{3}}{2}+3 x^{2}+6 x+\frac{9}{2}$
2. 1. Show that $f^{-1}$ exists.
2. f(x) can be written in the form $p(x+2)^{3}+q $, where p,$ q \in \mathbb{R}$ . Find the value of p and the value of q .
3. Hence, find $f^{-1}(x) $.
The graph of f(x) may be obtained by transforming the graph of y=x^{3} using a sequence of three transformations.
3. State each of the transformations in the order in which they are applied.
4. Sketch the graphs of y=f(x) and y=f^{-1}(x) on the same set of axes, indicating the points where each graph crosses the coordinate axes.
