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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 $$F(x,y,z):= x^2 +y^2 +z^2 -1 \qquad \text{and} \qquad G(x,y,z) :=x^3+y^3 + z^3.$$

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@user10938: Have you learnt method of Lagrange multipliers? – user17762 May 15 '11 at 17:54
up vote 4 down vote accepted

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.

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