 Utility conjugate

For an increasing concave utility on positive wealth, its utility conjugate uses the indicated supremum and is a convex function of the positive price variable. If $F(x)=-U(x)$ on positive $x$, extended by infinity elsewhere, then $\hat{U}(y)=F^{\ast }(-y)$. Under the Inada conditions its optimizer is the inverse marginal utility, and its derivative is the negative of that optimizer. The sign convention is part of the definition.