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Inequality of Jensen. Prove that if the function f(x) is convex upward on [a,b], then for any distinct points x1,x2,,xn (n2) from [a;b] and any positive α1,α2,,αn such that α1+α2++αn=1, the following inequality holds: f(α1x1++αnxn)>α1f(x1)++αnf(xn).

Prove that if the function f(x) is convex upwards on the line [a,b], then for any distinct points x1,x2 in [a;b] and for any positive α1,α2 such that α1+α2=1 the following inequality holds: f(α1x1+α2x2)>α1f(x1)+α2f(x2).