Concave supporting-tangent inequality ($U(y)-U(x)\le U^{\prime }(x)(y-x)$)

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.