<p>In general, FOCs produce local, not necessarily global, optima. Consider 
<span class="math-container">\begin{equation}
\max_x\;f(x)=x \sin(x) \quad\text{s.t. $x\in[0,7]$}.
\end{equation}</span></p>

<p>As the plot below shows, <span class="math-container">$f(x)$</span> has three stationary points in the domain <span class="math-container">$[0,7]$</span>: <span class="math-container">$x_1=0$</span>, <span class="math-container">$x_2\approx 2.03$</span>, and <span class="math-container">$x_3\approx4.91$</span>. Among them, <span class="math-container">$x_2$</span> is the local maximum that would be picked up by the FOC. However, the global maximum happens at the boundary, where <span class="math-container">$x=7$</span>. Since <span class="math-container">$f'(7)\ne0$</span>, the global maximum would be missed if you rely only on the FOC. </p>

<p><a href="https://i.sstatic.net/QQURN.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QQURN.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2019-02-21T15:27:14.597Z
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
CC BY-SA 4.0
2019-02-21T16:23:13.267Z
@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.]
Herr K.
CC BY-SA 4.0
2019-02-21T01:35:49.663Z
2019-02-21T01:35:49.663Z
Herr K.
<p>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?</p>

<mathematical-economics><optimization>
Constrained Optimization using Lagrangian method
Kautilya
# Constrained Optimization using Lagrangian method

(Asked by: username_0 on 2019-02-20)

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?

**Tags:** mathematical-economics, optimization

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## 1 answers

### Answer by username_1 on 2019-02-21. Score: 2

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](https://i.sstatic.net/QQURN.png)](https://i.sstatic.net/QQURN.png)

> **Comments:**
> **username_2**: The Lagragian would catch the boundary points, if we follow Kuhn and Tucker and consider the constraints $x \geq 0 $ and $x \leq 7$.  
> **username_1**: @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.]  

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