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- Jun 20, 2014

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Let $f : \Bbb R^n \to \Bbb R$ be differentiable at $\mathbf{x}_0$. If $f(\mathbf{x}_0) = 0$ and $\mathbf{x}_0$ has an open neighborhood $V \subset \Bbb R^n$ such that $f(\mathbf{x}) \ge 0$ for all $\mathbf{x}\in V$, prove that $\mathbf{x}_0$ is a critical point of $f$, i.e., $\nabla f(\mathbf{x}_0) = \mathbf{0}$.

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