Topological lower bounds in terms of Betti numbers for the number of critical points form a smooth function on a smooth manifold.
Morse Inequalities
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Weisstein, Eric W. "Morse Inequalities." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MorseInequalities.html