Consider the function $z = x^{5}y + 5 x^{4} - 1 y$. Find and classify all critical points of the function. If there are more blanks than critical points, leave the remaining entries blank. The critical point with the smallest $x$-coordinate is
(, ) Classification: local minimum local maximum saddle point cannot be determined (local minimum, local maximum, saddle point, cannot be determined)

The critical point with the next smallest $x$-coordinate is
(, ) Classification: local minimum local maximum saddle point cannot be determined (local minimum, local maximum, saddle point, cannot be determined)

The critical point with the next smallest $x$-coordinate is
(, ) Classification: local minimum local maximum saddle point cannot be determined (local minimum, local maximum, saddle point, cannot be determined)