There is a critical point when dz/dy and when dz/dx are equal to zero

dz/dx= 5yx^{5y-1}+ y^{5}+ y

This is zero at (0,0)

dz/dx= x^{5y}lnx^{5}+ 5xy^{4}+ 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