First order optimality conditions

Question

Let $X$ and $Y$ be two Banach spaces, and $f(x,y)$ be a nonlinear maps that maps $X\times Y$ to $\mathbb{R}$. The mapping $f$ is Frechet differentiable with respect to each variable $x$ and $y$. In order to obtain the first-order necessary optimality conditions, one needs to set $$f'_x=f'_y=0.$$ Do we need to check if the second order derivative $f'_{x,y}$ exists?