It's obvious that x=y=z=1 is an all-positive, all-real, solution of

x + y² + z³ = 3

y + z² + x³ = 3

z + x² + y³ = 3

Can you prove that it is the only such solution?

To get a zero sum===> not necesseraly forces all the parts

to be zero i.e. some of the " dx , dy ,dz "

might be negative ...AND SOME POSITIVE