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    Advanced Calculus: The Mean Value Theorem and Directional Derivatives

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    Let F: R^n --> R be continuously differentiable. Show that at each point x E R^n there is a direction hx so that the directional derivative is 0, i.e., df/dhx (x) = 0. Is hx unique? Give a method for determining hx.

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    Proof: Since is continuously differentiable, then exists. is defined as . Now for any ...

    Solution Summary

    A proof involving directional derivatives is provided. The solution is detailed.