Convex Function

Definition

$\forall 0\le t\le 1,\forall x_1,x_2\in X,f(t{x}_1+(1-t)x_2)\le tf(x_1)+(1-t)f(x_2)$

If $f$ is differentiable on $(a,b)$, then

• $\forall a<x<y<b,f^{\prime }(x_2)\le f^{\prime }(x_2)\iff f$ is convex
• $\forall a<x<b,f^{\prime \prime }(x)>0\iff f$ is convex