Problem 1292 (difficulty: 5/10)

Let \(\displaystyle f:\R^3\to\R\) be twice differentiable. Prove that if

\(\displaystyle \left<f'(x,y,z), (x,y,z)\right> \ge 0, \)

holds everywhere, then

\(\displaystyle D_{11}f(0,0,0) + D_{22}f(0,0,0) + D_{33}f(0,0,0) \ge 0. \)

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