Inada utility with vanishing curvature ($U(x)=\log x+x^{-1}-\frac{1}{6}{x}^{-2}$)

This smooth utility has $U^{\prime }(x)=x^{-1}-x^{-2}+x^{-3}/3>0$ and $U^{\prime \prime }(x)=-(x-1{)}^2/x^4$. Its derivative is strictly decreasing, diverges at zero and tends to zero at infinity, so the utility is strictly concave and satisfies the Inada conditions. Yet its curvature vanishes at one. The inverse marginal utility cannot have a finite derivative at $y=1/3$, since differentiating $U^{\prime }(I(y))=y$ there would give zero equal to one. Its dual is differentiable but not twice differentiable at that price.