maximum value of the expression : $2x+3y+z$ as $(x,y,z)$ varies over the sphere $x^2+y^2+z^2=1$ I have to find out the maximum value of the expression :


$2x+3y+z$ as $(x,y,z)$ varies over the sphere $x^2+y^2+z^2=1.$



Let $(x,y,z)=(\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta) \in S^2$
\begin{align*}
  2\cos \phi+3\sin \phi & \le \sqrt{2^2+3^2} \\
  2x+3y &= \sin \theta (2\cos \phi+3\sin \phi) \\
  & \le \sqrt{13} \, |\sin \theta| \\
  2x+3y+z & \le \sqrt{13} \, |\sin \theta|+\cos \theta \\
  & \le \sqrt{13+1} \\
  &= \sqrt{14}
\end{align*}
