Answer
$(\dfrac{5}{14},\dfrac{2}{7},\dfrac{19}{14})$
Work Step by Step
Use Lagrange Multipliers Method:
$\nabla f=\lambda \nabla g$
Need to find the equation of line that is orthogonal to the plane and passes through $(0,1,1)$.
Equation of plane is: $x-2y+3x=6$
Thus,
$0+(1)(p)-2(1-2(p))+3(1+3(p))=6$
After solving, we get $p=\dfrac{5}{14}$
Thus, $x=p,y=1-2p,z=1+3p$
Plug in the value of $p$, we get
$x=\dfrac{5}{14}$, $y=\dfrac{2}{7}$ and $z=\dfrac{19}{14}$
The required points are:
$(\dfrac{5}{14},\dfrac{2}{7},\dfrac{19}{14})$
