Concave supporting-tangent inequality
= Concave supporting-tangent inequality
{title2=$U(y)-U(x)\leq U'(x)(y-x)$}
For a differentiable <concave> function, secant slopes lie below the tangent slope at their left endpoint and above the tangent slope at their right endpoint. This gives the displayed inequality for either order of $x,y$. It converts a first-order optimality relation into a global utility bound. Strict <concavity> makes equality possible only at $x=y$, on the differentiability domain.