Solve this set of Lagrange multiplier equations,

Question

I'm trying to solve

$$(yz,xz, xy) = (\lambda\frac{2x}{a^2},\lambda\frac{2y}{b^2},\lambda\frac{2z}{c^2})$$

with the constraint equation

$$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$

What's the best way to proceed? I can't seem to first solve for $\lambda$.

Thanks,

Answer 1

We have: $3(xyz)^2 = 3xy\times yz \times zx = \dfrac{24xyz\times \lambda^3}{a^2b^2c^2}, 3xyz = xyz+xyz+xyz = 2\lambda\left(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}\right)=2\lambda \times 1 = 2\lambda\Rightarrow 2\lambda = 3xyz=\dfrac{24\lambda^3}{a^2b^2c^2}\Rightarrow \lambda^2 = \dfrac{a^2b^2c^2}{12}\Rightarrow \lambda = \dfrac{abc}{2\sqrt{3}}$.

Answer 2

If any of $x$, $y$, and $z$ is $0$, then two of them are $0$ and the other is not (so as to satisfy $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$). Thus, $\lambda=0$. Hence, all such solutions are $(x,y,z,\lambda)=(\pm a,0,0,0)$, $(x,y,z,\lambda)=(0,\pm b,0,0)$, and $(x,y,z,\lambda)=(0,0,\pm c,0)$.

If none of $x$, $y$, and $z$ is $0$, then $$\frac{8\lambda^3 (xyz)}{(abc)^2}=\left(\lambda\frac{2x}{a^2}\right)\left(\lambda\frac{2y}{b^2}\right)\left(\lambda\frac{2z}{c^2}\right)=(yz)(zx)(xy)=(xyz)^2\,.$$ Consequently, $\lambda=\frac{\sqrt[3]{(xyz)(abc)^2}}{2}$. Now, note that $\lambda \neq 0$, so $$\frac{x}{y}=\frac{\lambda\left(2x/a^2\right)}{\lambda\left(2y/b^2\right)}\left(\frac{a}{b}\right)^2=\frac{yz}{zx}\left(\frac{a}{b}\right)^2\,,$$ so $$\left(\frac{x}{a}\right)^2=\left(\frac{y}{b}\right)^2\,,\text{ or }\frac{|x|}{|a|}=\frac{|y|}{|b|}\,.$$ Similarly, $\frac{|y|}{b}=\frac{|z|}{c}$. That is, to comply with $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$, we must have $$\frac{|x|}{|a|}=\frac{|y|}{|b|}=\frac{|z|}{|c|}=\frac{1}{\sqrt{3}}\,.$$ With $(x,y,z)=\left(u\frac{a}{\sqrt{3}},v\frac{b}{\sqrt{3}},w\frac{c}{\sqrt{3}}\right)$ where $u,v,w\in\{-1,+1\}$, we get $\lambda=\frac{(uvw)(abc)}{2\sqrt{3}}$. Hence, there are $8$ solutions $$(x,y,z,\lambda)=\left(u\frac{a}{\sqrt{3}},v\frac{b}{\sqrt{3}},w\frac{c}{\sqrt{3}},\frac{(uvw)(abc)}{2\sqrt{3}}\right)\,,$$ where $u,v,w\in\{-1,+1\}$.

P.S. Are you trying to optimize $xyz$ given that $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$? There is a very simple AM-GM solution to this problem.