Strict concavity ($f(tx+(1-t)y)>tf(x)+(1-t)f(y)$)

Strict concavity is the displayed property for distinct $x,y$ in a convex set and $0<t<1$. It makes a stationary point of a differentiable function its unique global maximum, whenever such a stationary point exists.

A compact convex subset of ${\mathbb{R}}^2$ meeting the positive quadrant has a unique positive point maximizing $xy$. Existence follows by maximizing the continuous product on the intersection with the closed nonnegative quadrant; a positive attainable value excludes its axes. Uniqueness follows from strict concavity of $\log x+\log y$ on the positive quadrant. This is a useful way to optimize a product even though the product itself is not concave.