Finding Maximum Value of f(x,y,z) on Ellipsoid x^2+2y^2+3z^2=1

[yifli]

Homework Statement

Find the maximum value of $$f(x,y,z)=x+y+z$$ on the ellipsoid $$x^2+2y^2+3z^2=1$$

2. The attempt at a solution
Is it possible to find the maximum value without using Lagrange multiplier?

Thanks

 

[MaxManus]

deleted

 

[HallsofIvy]

You could solve
$$x^2+ 2y^2+3z^2= 1$$
for, say,
$$x=\pm\sqrt{1- 2y^2- 3z^2}$$
and then maximize
$$\sqrt{1- 2x^2- 3y^2}+ y+ z$$ and
$$-\sqrt{1- 2y^2- 3y^2}+ y+ z$$