Concave function

A function $f$ is concave exactly when $-f$ is convex. Equivalently,for $0\le t\le 1$.

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.

A concave majorant of $F$ is a concave function lying everywhere above $F$ on the specified domain. The least concave majorant, when it exists, is the pointwise smallest such function. A common tangent can replace an upward derivative kink by a linear segment joining two contact points.

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