# example of tests for local extrema in Lagrange multiplier method

Let $n\in\mathbb{N}^{+}$ and $c\in\mathbb{R}$. We want to find the local extrema of the function

 $f\colon\mathbb{R}^{n}\to\mathbb{R},\quad x\mapsto\sum_{1\leq i

subject to the condition $g=0$, where

 $g\colon\mathbb{R}^{n}\to\mathbb{R},\quad x\mapsto\sum_{1\leq i\leq n}x_{i}-c.$
 $\partial_{i}f(x)=\sum_{k\neq i}x_{k},\quad\partial_{i}g(x)=1,$
 $\partial_{i}\partial_{j}f(x)=1-\delta_{i,j},\quad\partial_{i}\partial_{j}g(x)=0,$

where $\delta_{i,j}$ is the Kroenecker-delta. Thus the necessary condition $f^{\prime}(x)=\lambda g^{\prime}(x)$ together with $g(x)=0$ gives the system of equations

 $\sum_{j\neq i}x_{j}=\lambda,\quad i\in\{1,\dots,n\},$
 $\sum_{1\leq j\leq n}x_{j}=c.$

By summing the first $n$ equations and then substituting in the last we get

 $(n-1)c=n\lambda,$
 $x_{i}=\sum_{1\leq j\leq n}x_{j}-\sum_{j\neq i}x_{j}=c-\lambda=\frac{c}{n},% \quad i\in\{1,\dots,n\}.$

Thus there is only one point, where local extremum is possible. We apply the test in the parent entry to the matrix

 $D^{2}(f-\lambda g)(x)=[1-\delta_{i,j}]_{i,j=1}^{n}=nP-I,$

where $P$ is the matrix containing $1/n$ in all entries, and $I$ is the identity matrix  . $P$ is a rank one projection. Therefore the second derivative has spectrum $\sigma(nP-I)=\{n-1,-1\}$, where $-1$ has multiplicity $n-1$, and $n-1$ has multiplicity $1$. Thus the second derivative of $f-\lambda g$ is indefinit, so it has no local extrema. However the nullspace  of $g^{\prime}(x)$ is precisely the nullspace of $P$, thus the second derivative is strictly negative on the tangent space $T_{x}(M)$, so the vector $(c/n,\dots,c/n)$ is a local maximum of $f$ subject to $g=0$.

Title example of tests for local extrema in Lagrange multiplier method ExampleOfTestsForLocalExtremaInLagrangeMultiplierMethod 2013-03-22 19:12:19 2013-03-22 19:12:19 scineram (4030) scineram (4030) 9 scineram (4030) Example msc 26B12 msc 49K35 msc 49-00