# Math Help - Lagrange Multplier

1. ## Lagrange Multplier

I have to maximise $2 \tan ^{-1} (x_{1}) + x_{2}$ subject to $x_{1} + x_{2} \le b_{1}$ and $- \log (x_{2}) \le b_{2}$ for constants $b_{i}$ satisfying $b_{1} - e^{-b_{2}} \ge 0$ where $x_{i} \ge 0$.

Clearly it is easy to write down the Lagrangian and get that $L_{x_{1}} = \frac{2}{1 + x_{1} ^{2} } - \lambda = 0$ and $L_{x_{2}} = 1 - \lambda + \frac{ \mu }{x_{2}} = 0$.

Note that slacked variables have been used to take care of the inequality constraints. I don't know how to proceed. Clearly I could solve the equations I have from differentiation however taking into account that I have inequalities for constraints I am rather confused. What can I do here?

2. Anyone?

3. I hate to bump this twice but I am still stuck. Does anyone have any ideas, even if you can't get the full answer?