How to Find Extrema of Linear Functions on a Sphere?

[jschmid2]

So, I seem to be drawing some blanks - my calc III is a little rusty.

Find the max(2x+2y+z) and min(2x+y+z) on the surface x²+y²+z²=1

At first I thought I would take partial derivatives, but none of them yield 0, so that's not going to work. Any suggestions would be mighty helpful because I will be dealing with problems a little harder than this, but if I can get this fundamental figured out, it will be extremely helpful.

Thanks.

 

[LCKurtz]

Try LaGrange multipliers.

 

[jschmid2]

That is exactly what I needed. Thank you so much for the hint! :)