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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?

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In general, FOCs produce local, not necessarily global, optima. Consider \begin{equation} \max_x\;f(x)=x \sin(x) \quad\text{s.t. $x\in[0,7]$}. \end{equation}

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.

enter image description here

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    $\begingroup$ The Lagragian would catch the boundary points, if we follow Kuhn and Tucker and consider the constraints $x \geq 0 $ and $x \leq 7$. $\endgroup$ – Bertrand Feb 21 '19 at 15:27
  • $\begingroup$ @Bertrand: The OP specifically asked about "[t]he stationary points that we derive by solving the first order conditions of the Lagrangian". [Emphasis is mine.] $\endgroup$ – Herr K. Feb 21 '19 at 16:23

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