= Solution
Take \b[$\mathbb{CP}^3$]. It is a closed <smooth manifold> and is <simply connected>: its <CW complex> structure has no one-cells, and attaching cells of dimension at least two to a point cannot create a <fundamental group>. Its <cohomology ring> is $\mathbb Z[x]/(x^4)$ with $|x|=2$, so $x^3\ne0$.
The theorem used is the <Lusternik–Schnirelmann critical point theorem>: every smooth real-valued function on a closed connected <smooth manifold> has at least its unnormalized <Lusternik–Schnirelmann category> many distinct <critical points>, without a nondegeneracy assumption. Here category is the minimum number of open subsets covering the manifold whose inclusions are <null-homotopic>. The <cup length> bound gives category at least four. To justify that bound, a cover by three categorical sets would make each positive-degree class vanish on its assigned set. The <long exact sequence> of each pair lifts three copies of $x$ to relative classes, and their <relative cup product> would lie in $H^6(M,M;\mathbb Z)=0$, contradicting $x^3\ne0$. Thus \b[every smooth function has at least four critical points], including functions with degenerate critical points. The critical-point theorem in this formulation is described in https://doi.org/10.1016/j.exmath.2017.10.003[Conley pairs in geometry—Lusternik–Schnirelmann theory and more].
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