Finding extremum of a function using Lagrange

Question

$f(x, y) = \ln(x) + \ln(y) $ Restricted to $g(x, y) = x + y/2 - 1 = 0$

I did $\delta f(x, y) = \lambda \delta g(x, y) $

I have the system $1/x = \lambda$ and $1/y = \lambda /2$

But I'm stuck here. How to find lambda to find the critical points? Any help please

Answer 1

Your system is incomplete. It should be$$\left\{\begin{array}{l}\frac1x=\lambda\\\frac1y=\frac\lambda2\\x+\frac y2-1=0.\end{array}\right.$$Its only solution is $(x,y,\lambda)=\left(\frac12,1,2\right)$.

Answer 2

There's definitely no need for Lagrange. As $x=1-y/2$ we have $$ \ln(x)+\ln(y)=\ln(xy)=\ln\bigl((1-y/2)y\bigr). $$ As $\ln$ is strictly monotone increasing and $(1-y/2)y$ takes it maximum for $y=1$ we're done: $x=1/2$ and the maximal value of $f$ is $\ln(1/2)$.