Question 7
What can we conclude from the first order conditions
\(x_{2}-\lambda p_{1}=0\) and \(x_{1}-\lambda p_{2}=0\)?
True: The function
\(L\left(x_{1},x_{2}\right)=\log\left(x_{1}\right)+log(x_{2})-\lambda\left(p_{1}x_{1}+p_{2}x_{2}\right))\)
has a unique local extremum given by
\(\left(x_{1},x_{2}\right)=(\lambda p_{2},\lambda p_{1})\)
False: The function
\(L\left(x_{1},x_{2}\right)=\log\left(x_{1}\right)+log(x_{2})-\lambda\left(p_{1}x_{1}+p_{2}x_{2}\right))\)
has a unique local extremum given by
\(\left(x_{1},x_{2}\right)=(p_{1},p_{2})\)