Is it possible for a real $C^2$ function $f:U \subset \mathbb{R}^n \rightarrow \mathbb{R} $ with $U$ an open set, to have a point $p \in U$ such that the second partial derivative $f_{11}(p) = 0$ but $f_1(p) \neq 0$? In the case of one dimension, this would translate to $f''(p) = 0$ but $f'(p) \neq 0.$
I believe this to be true, but I cannot come about proving by the definition of the second derivative; I believe that taking the limit as the increment goes to zero should make me conclude that we are forced to have $f'(x) = 0,$ but I am not certain of this.
Thanks in advance!