The paragraph is given below:
But I do not understand:
1-In the forth line why we can not have the case $df_{x} =$ constant other than 0, could anyone explain this for me please?
2-In the sixth line how f is simply the first coordinate function, could anyone give me a concrete example for describing this please?
3-In the tenth line I could not understand why the authors said "But if $f(x)$ ia an extreme value, then obviously $f$ can not be a coordinate function near x" , could anyone explain this statement for me please may be by a concrete example?
thank!
(1) The only constant linear map is the zero map. (2) The claim in the sixth line is essentially the implicit function theorem. (3) Consider $f(x)=x^2$. No change of coordinates will turn this into $x$ near $0$.