I have recently been tackling the following problem:
If $a+b+c = 0 $ and $ a^2 + b^2 +c^2 = 1$, work out $a^4 +b^4 +c^4$.
Could this problem admit a solution through the method of lagrange multipliers. Is my thinking in that the maximum and minimum are the same, and thus the method is valid, correct or flawed? If the latter is true, could you please point out the error in my reasoning, otherwise show how to carry out the method, with full working. I have already solved the problem using nothing more than basic algebraic manipulation, but it would be interesting to see if this method works.
Just for info (see comments), let $a,b,c$ be the roots of a cubic polynomial.
Since $a+b+c=0$ this has form $x^3+px+q=0$ where $$p=ab+bc+ca=\cfrac {(a+b+c)^2-(a^2+b^2+c^2)}2=-\cfrac 12$$
Now note multiply by $x$ to see that $a,b,c$ satisfy $f(x)=x^4+px^2+qx=0$
Then $$0=f(a)+f(b)+f(c)=(a^4+b^4+c^4)+p(a^2+b^2+c^2)+q(a+b+c)$$
Substituting known values gives $a^4+b^4+c^4=\cfrac 12$
Working with polynomials is sometimes a convenient way of capturing and organising information about symmetric functions.