# Constrained Optimization using Lagrangian method

The stationary points that we derive by solving the first order conditions of the Lagrangian are those points global optimum points or local optimum points?

In general, FOCs produce local, not necessarily global, optima. Consider $$$$\max_x\;f(x)=x \sin(x) \quad\text{s.t. x\in[0,7]}.$$$$
As the plot below shows, $$f(x)$$ has three stationary points in the domain $$[0,7]$$: $$x_1=0$$, $$x_2\approx 2.03$$, and $$x_3\approx4.91$$. Among them, $$x_2$$ is the local maximum that would be picked up by the FOC. However, the global maximum happens at the boundary, where $$x=7$$. Since $$f'(7)\ne0$$, the global maximum would be missed if you rely only on the FOC.
• The Lagragian would catch the boundary points, if we follow Kuhn and Tucker and consider the constraints $x \geq 0$ and $x \leq 7$. – Bertrand Feb 21 '19 at 15:27