# Hessian matrix in $P_0$ when partial derivatives are zero

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.

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No. You should differentiate for the second time before evaluation, and the result of evaluation may well be nonzero. For a concrete example take $U(x,y)=x^2+xy+y^2$ at $P=0$. All first order derivatives vanish at P, but the second order derivatives are nonzero.