If $$ f_x=f_y=f_{xx}=f_{yy}=1$$ at a point, does it imply $$f_{xy}=f_{yx}=1 $$ at that point.
Note this is not a homework problem, it is a question from a book I am reading for which the solution just regards $f_{xy}=f_{yx}=1$ given $ f_x=f_y=f_{xx}=f_{yy}=1$ (Thomas Calculus 12th edition page 832 question 96). So I was wondering if there was a mistake in the question which is, simply forgetting to add $f_{xy}=f_{yx}=1 $ as a premise.