Before revision:  The existence follows from an algorithm that converges from any starting point to such a point in a number of gradient and function evaluations f(x,y), and \nabla^2_y f(x,y), } that is polynomial in \frac{1{\varepsilon}}  , the dimension d, and the bounds on f and its first three derivatives.
After revision:  The existence follows from an algorithm that converges from any starting point to such a point in a number of evaluations to f, \nabla_{y f(x,y), and \nabla^2_y f(x,y), } that is polynomial in \frac{1{\varepsilon}} , the dimension d, and the bounds on f and its first three derivatives.