Dual differentiability with nonvanishing utility curvature (${\hat{U}}^{\prime }(y)=-I(y),{\hat{U}}^{\prime \prime }(y)=-1/U^{\prime \prime }(I(y))$)

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