I take a $U(x,y)$ on $\mathbb{R}^2$, then a fixed $P_0=(x_0,y_0)$ where $U_x(P_0)=U_y(P_0)=0$.
Can I automatically say that $U_{xx}(P_0)=U_{xy}(P_0)=U_{yy}(P_0)=0$ (without calculating before $U_{xx}$, $U_{xy}$, $U_{yy}$ and then evaluating them on $P_0$)?
Thanks a lot.