I would appreciate if somebody could help me with the following problem:
Find the maximum of the function $$f(x,y,z) = x$$ on the curve defined by the equations $F(x,y,z) = G(x,y,z) =0$ with $$F(x,y,z):= x^2 +y^2 +z^2 -1 \qquad \text{and} \qquad G(x,y,z) :=x^3+y^3 + z^3.$$