Strictly concave function

A function is strictly concave when its concavity inequality is strict for distinct points and coefficients strictly between zero and one. A strictly concave function has at most one maximizer on a convex set.

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.

If a twice differentiable real function on $\mathbb{R}$ satisfies $f^{\prime \prime }(x)\le -c$ for some fixed $c>0$, then it has a unique global maximum. The mean value theorem shows that $f^{\prime }$ is strictly decreasing, and that $f^{\prime }(x)+cx$ is nonincreasing. Hence $f^{\prime }(x)\le f^{\prime }(0)-cx$ for $x>0$ and $f^{\prime }(x)\ge f^{\prime }(0)-cx$ for $x<0$. These inequalities force opposite derivative signs at the two ends. The intermediate value theorem supplies a unique zero of $f^{\prime }$, at which $f$ changes from increasing to decreasing. Merely having $f^{\prime \prime }<0$ does not guarantee existence: $f(x)=-e^{-x}$ is increasing and approaches its unattained supremum zero.

On the positive orthant,is strictly concave. Its unique maximizer $x^{\ast }$ on a compact convex feasible set satisfies the first-order inequalityequivalently