In the text, it says:
Consider smooth function on a manifold $f: X \to \mathbb{R}$. If $f(x)$ is an extreme value, then $f$ cannot be a coordinate function near $x$, so $df_x$ must be zero.
I don't understand here - if $f(x)$ is an extreme value, why $f$ cannot be a coordinate function?
Writing $f$ in terms of some coordinate system in a neighborhood of $f$, we can reformulate the question as: if $F=(f_1,\dots,f_n)$ is a diffeomorphism between open subsets of $\mathbb R^n$, can $f_1$ have a point of local extremum at $x$? The answer is no, for otherwise $\nabla f_1(x)$ would vanish, making $\det DF(x)=0$.