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: \(f(\alpha_1 x_1 + \alpha_2 x_2 ) > \alpha_1 f (x_1) + \alpha_2 f(x_2)\).
Inequality of Jensen. Prove that if the function \(f (x)\) is convex upward on \([a, b]\), then for any distinct points \(x_1, x_2, \dots , x_n\) (\(n \geq 2\)) from \([a; b]\) and any positive \(\alpha_{1}, \alpha_{2}, \dots , \alpha_{n}\) such that \(\alpha_ {1} + \alpha_{2} + \dots + \alpha_{n} = 1\), the following inequality holds: \(f (\alpha_{1} x_1 + \dots + \alpha_{n} x_n) > \alpha_{1} f (x_1) + \dots + \alpha_{n} f (x_n)\).