Problems

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: $f({\alpha }_1{x}_1+\cdots +{\alpha }_n{x}_n)>{\alpha }_{1}f(x_1)+\cdots +{\alpha }_{n}f(x_n)$.

Prove that if the function $f(x)$ is convex upwards on the line $[a,b]$, then for any distinct points $x_1,x_2$ in $[a;b]$ and for any positive ${\alpha }_1,{\alpha }_2$ such that ${\alpha }_1+{\alpha }_2=1$ the following inequality holds: $f({\alpha }_1{x}_1+{\alpha }_2{x}_2)>{\alpha }_{1}f(x_1)+{\alpha }_{2}f(x_2)$.