Existence of critical points

Question

I want to prove:

We denote the $n$-sphere by $S^n$. Let $f : S^2 \to S^1$ be a $C^{\infty}$ function. Then $f$ has at least two critical points on $S^2$.

My effort: Let $p : \mathbb{R} \to S^1$ be a covering map and assume $f(y_0) = x_0$ and $p(\tilde{x}_0) = x_0$. Then $f_*(\pi_1(S^2)) \subset p_*(\pi(\mathbb{R},\tilde{x}_0 ))$. Then there is a lift $\tilde{f} : S^2 \to \mathbb{R}$ such that $p \circ \tilde{f} = f$. Thus $\tilde{f}$ has a maximum value and a minimum value. So $\tilde{f} $ and $f$ have at least two critical points.

Question: I am not sure my proof is correct.

Answer 1

Hint: Consider the restriction $g:C^1\rightarrow S^1$ of $f$ to a big circle contained in $S^2$. It is not surjective. To see this remark that a big circle retracts to a point in $S^2$ (or it defines an element of the fundamental group of $S^2$ which is trivial and the image of this element by $f$ is necessarily trivial.)

Let $x$ which is not in the image of $g$, and $P_x$ the stereographic projection centered at $x$, $P_x\circ g: C^1\rightarrow \mathbb{R}$ is a diiferentiable map which has a minimum and a maximum, show that the extrema points are critical points.