I'm trying to work out the following problem for my microeconomics course, using the standard method we have been taught.
Problem: Output is a function of inputs $x_1$ and $x_2$, $y=x_1^{1/2}x_2$. Total cost is a function of inputs $x_1$ and $x_2$ and their corresponding prices, $w_1=3$ and $w_2=2$, $c=3x_1+2x_2$. The price at which output can be sold is $p=6$. What is the profit maximizing level of $x_2$ for the firm to use in the long run?
Attempt: In the long run, the profit maximizing level of $x_2$ is found by solving $p\times MP_2=w_2$. That is, $6x_1^{1/2}=2$. But, $x_2$ is no longer a variable in this equation! This leads me to believe the firm will exit, meaning $x_2=0$, but I cannot think of how to relate the economic theory to the mathematics.