Dimension of intersection of two manifold

Question

For any $f\in C^\infty(X)$, $X$ smooth manifold. Define $$X_{df}:=\{(x,df_x): x\in X, df_x= T^*_x X\}$$ $$X_0:=\{(x,\zeta): \zeta=0 \text{ in } T_x^*X\}$$

In the exercise we are asked for proof: If $X$ is compact, $$\#\{X_{df}\cap X_0\}\geq 2$$

Answer 1

Hint: Global max and global min.