How to show that there are 2 points of $S^n$ where a smooth map $g: S^n \to\mathbb R$ has tangent map equal to zero?
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$S^n$ is compact, so on a purely topological level, $g$ (as a continuous function on it) must have a maximum and a minimum. Now if $g$ is also smooth, these must be critical values, by the same argument as in elementary calculus. Since the range of $g$ is 1-dimensional, anywhere $dg_x$ does not have full rank it must be zero.