<s>[INST] Optimization of $2x+3y+z$ under the constraint $x^2+ y^2+ z^2= 1$ 
Let $x, y$ and $z$ be real numbers such that $x^2+ y^2+ z^2= 1$. Find the
  maximum and minimum values of $2x + 3y + z$ .    

How can I able to solve the problem? Thanks for your help.
 [/INST] Using the Cauchy-Schwarz inequality, $2x+3y+z\leq \sqrt{2^2+3^2+1^2}\sqrt{x^2+y^2+z^2}=\sqrt{14}$ which is the maximum value.Since $x$,$y$ and $z$ are reals, the minimum value occurs when $x$,$y$ and $z$ are negative which can be used to infer that the minimum value is $-\sqrt{14}$.
</s>