I am having trouble with this problem... I need to find the maximum distance from the origin to the surface $$f:=\frac{x^4}{2^4}+\frac{y^4}{3^4}+\frac{z^4}{(\sqrt2)^4}-1$$
I think I have to use Lagrange multipliers, and form the function: $$L:=x^2+y^2+z^2 - \lambda(\frac{x^4}{2^4}+\frac{y^4}{3^4}+\frac{z^4}{(\sqrt2)^4}-1) $$
Next I would do partial derivatives: $$Lx:=2x-\frac{1}{4}\lambda x^3 $$ $$Ly:=2y-\frac{4}{81}\lambda y^3 $$ $$Lz:=2z-\frac{4\lambda z^3}{(\sqrt2)^4} $$ $$L\lambda:=1-(\frac{1}{16})x^4-(\frac{1}{81})y^4-\frac{z^4}{(\sqrt2)^4} $$ And solving the system I got the points:
- $(\frac{2\sqrt2}{\sqrt\lambda},\frac{9\sqrt2}{2\sqrt\lambda},\frac{\sqrt2}{\sqrt\lambda})$
- $(-\frac{2\sqrt2}{\sqrt\lambda},-\frac{9\sqrt2}{2\sqrt\lambda},-\frac{\sqrt2}{\sqrt\lambda})$
If I replace on $x^2+y^2+z^2$ I have a maximum distance: $\sqrt101$
Am I right? Could anybody help me? I have a problem plotting the function f with Maple. I am writing: $$plot3d(f(x, y, z), x = -1 .. 1, y = -1 .. 1, z = -1 .. 1)$$ But it gives me the error "Error, (in plot3d) unexpected option: z = -1 .. 1 "
Thank you. Sorry for my english, is not my native language.