Trying to tackling a convexity problem. Let $f(x)$ be a real-valued differentiable function on $R^n$. If $f(x)$ is strictly convex, prove that $\nabla f(x) = \nabla f(y)$ if and only if $x = y$.
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1This is equivalent to the same property for $n = 1$, since it will be true of $f$ restricted to the line through $x$ and $y$. – David Feb 01 '16 at 17:54