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