TITLE: Find the maximum
QUESTION [5 upvotes]: 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 
$$F(x,y,z):= x^2 +y^2 +z^2 -1 \qquad \text{and} \qquad G(x,y,z) :=x^3+y^3 + z^3.$$

REPLY [4 votes]: Using the method of Lagrange multipliers should yield the following system of equations:
$\begin{eqnarray}
1&=&2ax+3bx^2\\
0&=&2ay+3by^2\\
0&=&2az+3bz^2\\
1&=&x^2+y^2+z^2\\
0&=&x^3+y^3+z^3\\
\end{eqnarray}$
Solving the second and third of these tells you that $y$ is either $0$ or $-2a/3b$ and $z$ is either $0$ or $-2a/3b$. This gives you four cases to work through.