Find the maximum and minimum values of f(x, y,z)=3x+3y+3zf(x, y,z)=3x+3y+3z on the sphere x2+y2+z2=1x2+y2+z2=1.

Lagrangian:



Set partial derivative equal to 0 and look for critical points:









We require , so we find that










which means there are two possible critical points, and . It's clear enough that the first gives a maximum value of , and a minimum value of .

Where are the functions?