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Found problems: 1

2002 IMC, 12
Tags: inequalities , gradient , real analysis
Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a convex function whose gradient $\nabla f$ exists at every point of $\mathbb{R}^{n}$ and satisfies the condition $$\exists L>0\; \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||\leq L||x_{1}-x_{2}||.$$ Prove that $$ \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||^{2}\leq L\langle\nabla f(x_{1})-\nabla f(x_{2}), x_{1}-x_{2}\rangle. $$