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

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