While studying for exams, I stumbled upon the following exercise.
Find the maxima and minima of $$f(x,y,z)=x^4+y^4+z^4$$ on the area given by $$\{(x,y,z): x^2+y^2+z^2=1, x+y+z=0\}$$
I am trying to use Lagrange's method of multiplication for two constraints, given in the definition of area, that is:
$$\nabla f(x,y,z)=\lambda\nabla g_1(x,y,z)+\mu \nabla g_2(x,y,z)$$
where $g_1$ is function defining sphere, and $g_2$ is the other. After some calculations I menaged to get a solution for $\lambda$ which is 1 in case $x\neq y\neq z$ or $x=y=z$ and for that case I've got some points. But now I have problem to get value of $\mu$ and solutions for $x,y,z$ assuming they are pairwise diffrent. In that case I've menaged to simplify all the equations to this problem: $$-4(x^2y+xy^2)=\mu$$ $$x^2+xy+y^2=1/2$$ Which I have no idea how to solve. Ofcourse I could have gone a wrong path at some point, as It's about my eight approach to this task. I've resarched about 10 sites on the web, as well as my lecture notes, books and other sources, and they proven themselves lacking in such examples, as often cases given by them are easly reductable, to the point, when some differences of parameteres and variables are equal to zero, or straightforwardly given. So because of lack of time, and fact that such examples are given in old exams, I'd appriciate some kind of detailed solution, or hint how to deal with them. Many thanks to anyone who have some time and will to help.
PS: English is not my main language, so sorry for grammar mistakes.
Eliminate $z$ from the constraints gives \begin{eqnarray*} x^2+xy+y^2=\frac{1}{2} \end{eqnarray*} as you noted in the question. Now we are trying to optimise \begin{eqnarray*} L=x^4+y^4+z^4=x^4+y^4+(x+y)^4=2(x^4+2x^3y+3x^2y^2+2xy^3+y^4) \\=2(x^2+xy+y^2)^2=\frac{1}{2}. \end{eqnarray*}