Use lagrange multultipliers to find the indicated extrema

Question

maximize $f(x,y,z)=x+y+z$ subject to $x^2+y^2+z^2=1$

I do not understand this at all or where to go from here would appreciate some insight

Answer 1

Let $g(x,y,z)=x^2+y^2+z^2$. Now, you need to solve the system $\nabla f(x,y,z)=\lambda \nabla g(x,y,z)$ and $g(x,y,z)=1$. That's the next step in Lagrange multipliers.

More precisely, use $\nabla f(x,y,z)=\lambda \nabla g(x,y,z)$ to solve for $x$, $y$, and $z$ in terms of $\lambda$ and then plug those formulae into $g$ and solve for $\lambda$ (Caution, you will need to address the case where $\lambda=0$ separately).