Home Competitions IMC 2002 Problem 12
Let f : R n → R f:\mathbb{R}^{n}\rightarrow \mathbb{R} f : R n → R ∇ f \nabla f ∇ f R n \mathbb{R}^{n} R n ∃ L > 0 ∀ x 1 , x 2 ∈ R n : ∣ ∣ ∇ f ( x 1 ) − ∇ f ( x 2 ) ∣ ∣ ≤ L ∣ ∣ x 1 − x 2 ∣ ∣ . \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}||. ∃ L > 0∀ x 1 , x 2 ∈ R n : ∣∣∇ f ( x 1 ) − ∇ f ( x 2 ) ∣∣ ≤ L ∣∣ x 1 − x 2 ∣∣. ∀ x 1 , x 2 ∈ R n : ∣ ∣ ∇ f ( x 1 ) − ∇ f ( x 2 ) ∣ ∣ 2 ≤ L ⟨ ∇ f ( x 1 ) − ∇ f ( x 2 ) , x 1 − x 2 ⟩ . \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. ∀ x 1 , x 2 ∈ R n : ∣∣∇ f ( x 1 ) − ∇ f ( x 2 ) ∣ ∣ 2 ≤ L ⟨ ∇ f ( x 1 ) − ∇ f ( x 2 ) , x 1 − x 2 ⟩ .
Dataset text AI-MO AoPS . Quoted with source.
Official text: International Mathematics Competition for University Students 2002 official paper , page 4.
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