Problem #PRU-61407

Problem

Inequality of Jensen. Prove that if the function $f(x)$ is convex upward on $[a,b]$, then for any distinct points $x_1,x_2,\dots ,x_n$ ($n\ge 2$) from $[a;b]$ and any positive ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n$ such that ${\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n=1$, the following inequality holds: $f({\alpha }_1{x}_1+\cdots +{\alpha }_n{x}_n)>{\alpha }_{1}f(x_1)+\cdots +{\alpha }_{n}f(x_n)$.