Utility function that generates a demand curve which will have an U shaped MR curve This is based off an answer given by @Ubiquitous in here: Can marginal revenue be increasing?

The solution he proposed involved a MR curve that sloped down, then up and then down. His equation for the demand curve (for example) to generate such a MR is $P(q)=1−q+3q^2−3q^3$ (see comments below his answer).

My question is: 

Is there an Utility function that can generate a demand curve which gives, say an U-shape looking MR?

Note: I am not looking for an utility function that replicates Ubiquitous's MR.
Let $v(x) = \int  1 − x + 3x^2 − 3x^3 \text{d}x$.
The quasi linear utility function
$$
U(x,y) = v(x) + y
$$
generates the inverse demand curve given by Ubiquitous if income $I$ is large enough and $p_y = 1$, as
\begin{align*}
|MRS(x,y)| &amp; = p_x \\
\\
v'(x) &amp; = p_x \\
\\
1 − x + 3x^2 − 3x^3 &amp; = p_x.
\end{align*}
Hence it will also result in the same $MR$ for $x$.

P.s.: I have not checked if $U(x,y)$ is convex, so second order conditions may need to be examined here. Note that $U(x,y)$ need not be convex everywhere, only above the $MR$'s U shape. Everywhere else it can be altered as we see fit.