Furthest along a direction

Idea. The expression is a dot product with a fixed vector. Over unit vectors it is largest when the unit vector points in the same direction as the fixed vector.

Step 1. Write $a=(1,2,3)$ and $u=(x,y,z)$ with $\Vert u\Vert =1$. Then $x+2y+3z=a\cdot u$.

Step 2: Cauchy-Schwarz. $$a\cdot u\le \Vert a\Vert \Vert u\Vert =\sqrt{1+4+9}=\sqrt{14},$$ with equality when $u$ points along $a$: $u=\frac{1}{\sqrt{14}}(1,2,3)$.

So the maximum is $\sqrt{14}\approx 3.742$, and the minimum is $-\sqrt{14}$.

With Lagrange multipliers. Setting the gradient of $x+2y+3z$ proportional to that of $x^2+y^2+z^2$ gives $(1,2,3)=2\mu (x,y,z)$: the same direction, and the same answer.