= Dual differentiability with nonvanishing utility curvature
{title2=$\widehat U'(y)=-I(y),\quad\widehat U''(y)=-1/U''(I(y))$}
For an increasing <strictly concave> differentiable utility satisfying <Inada conditions>, its utility conjugate has a unique optimizer $I(y)=(U')^{-1}(y)$ and <derivative> $-I(y)$. The dual is strictly decreasing and <strictly convex>. Twice <differentiability> of the dual additionally follows when $U''$ exists and is strictly negative everywhere. <Strict concavity> alone allows $U''$ to vanish and does not imply this additional assertion.
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