Can someone please verify an answer for me for the following Lagrange multiplier problem?
Consider the function in three variables defined by
$$ f(x,y,z) = xy + yz$$
and subject to the constraints \begin{align} x + 2y - 6 &= 0, \\ x - 3z &= 0. \end{align}
Find the critical point for the constrained problem using Lagrange multiplier method.
Would really appreciate if someone can just post the answer for this.
I really need to verify my answer.
HINT: consider the function $$L(x,y,z,\alpha,\beta)=xy+yz+\alpha(x+2y-6)+\beta(x-3z)$$ we get by differentiating $$f_x=y+\alpha+\beta=0$$ $$f_y=x+z+2\alpha=0$$ $$f_z=y-3\beta=0$$ $$f_{\alpha}=x+2y-6=0$$ $$f_{\beta}=x-3z=0$$ and we get $$\alpha=-2,\beta=\frac{1}{2},x=3,y=\frac{3}{2},z=1$$