Answer
$(0,3,0), (0,-3,0)$
Work Step by Step
Given: Equation of surface: $y^2=9+xz$
Need to apply Lagrange Multipliers Method to determine the points on the surface $y^2=9+xz$ that are closet to the $(0,0,0)$.
we have $\nabla f=\lambda \nabla g$
The closet distance can be found as:
$f(x,y)=D^2=x^2+y^2+z^2$
which gives $f_x=2x+z, f_z=x+2z$
Simplify to get the values for $x$ and $z$.
we found $x=0$ and $z=0$ and $y^2=9+xz=9+(0)(0)=9$
or, $ y=\pm 3$
Hence, our result is: $(0,3,0), (0,-3,0)$
