I was trying to find maxima/minima for $z= x^3 + y^3$ , subject to $x^2 + y^2 + z^2 = 1$. I tried to find it by using $f(x,y,z) = x^3 + y^3$ and $g(x,y,z) = x^2 + y^2 + z^2$ & by using the constraint as $g(x,y,z) = 1$. However that didn't workout , the way as expected. What am I missing here? How should $f$ & $g$ be set up to calculate Lagrange multipliers correctly?
$$\begin{align}f(x,y,z)&=z\\g(x,y,z)&=x^2+y^2+z^2-1=0\\h(x,y,z)&=x^3+y^3-z=0\end{align}$$ Then take the derivatives of $f+\lambda g+\mu h$: $$0+2\lambda x+3\mu x^2=0\\0+2\lambda y+3\mu y^2=0\\1+2\lambda z-\mu =0$$ We can rewrite these as $$x(2\lambda+3\mu x)=0\\y(2\lambda+3\mu y)=0 \\z=\frac{\mu-1}{2\lambda}$$ Obviously $x$ and $y$ cannot be simultaneously $0$, since it would imply $z=0$ in $h=0$, but $(0,0,0)$ does not verify $g=0$. Then you have three cases you need to solve (1) $x=0$, (2) $y=0$, and (3) $x=y\ne 0$.