There is a critical point when dz/dy and when dz/dx are equal to zero
dz/dx= 5yx5y-1+ y5+ y
This is zero at (0,0)
dz/dx= x5ylnx5+ 5xy4+ x
This is not defined at (0,0) because ln0 is not defined.
I do not know how to find the critical point but (0,0) is not it