points on surface that is closest to the point I need to Find the maximum and minimum values of  $$f(x, y, z) = x + 5y + 2z$$ on the sphere  $$x^{2} + y^{2} + z^{2} = 1$$

How about using Lagrange multipliers, as T.Bongers mentioned? You get this system of equations to solve for the extrema:
$
\begin{align*}
\quad\frac{\partial}{\partial x}\left( x + 5y + 2z - \lambda(x^2+y^2+z^2-1)\right)=0\\
\quad\frac{\partial}{\partial y}\left( x + 5y + 2z - \lambda(x^2+y^2+z^2-1)\right)=0\\
\quad\frac{\partial}{\partial z}\left( x + 5y + 2z - \lambda(x^2+y^2+z^2-1)\right)=0\\
\quad\frac{\partial}{\partial \lambda}\left( x + 5y + 2z - \lambda(x^2+y^2+z^2-1)\right)=0\\
\end{align*}
$
This becomes
$
\begin{align*}
&\quad1-2x\lambda=0\\
&\quad5-2y\lambda=0\\
&\quad2-2z\lambda=0\\
&\quad x^2+y^2+z^2-1=0\\
\end{align*}
$
and the maximum and minimum are at the $(x,y,z)$ points that solve the system: $\left(\pm\frac{1}{\sqrt{30}},\pm\frac{5}{\sqrt{30}},\pm\frac{2}{\sqrt{30}}\right)$, which André found another way.
