IMC 2002 · Problem 6

Let $f : \mathbb{R}^n \to \mathbb{R}$ be a convex function whose gradient $\nabla f = \left( \dfrac{\partial f}{\partial x_1}, \dots, \dfrac{\partial f}{\partial x_n} \right)$ exists at every point of $\mathbb{R}^n$ and satisfies the condition

$$\exists L > 0 \quad \forall x_1, x_2 \in \mathbb{R}^n \quad \|\nabla f(x_1) - \nabla f(x_2)\| \le L\|x_1 - x_2\|.$$

Prove that

$$\forall x_1, x_2 \in \mathbb{R}^n \quad \|\nabla f(x_1) - \nabla f(x_2)\|^2 \le L \langle \nabla f(x_1) - \nabla f(x_2), x_1 - x_2 \rangle. \tag{1}$$

In this formula $\langle a, b \rangle$ denotes the scalar product of the vectors $a$ and $b$.