Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
mymathcourses
Group Title
find the maximum and minumum values of the n variable function:X1+X2+...Xn subject to the constraint X1^2+X2^2+...+Xn^2=1
 2 years ago
 2 years ago
mymathcourses Group Title
find the maximum and minumum values of the n variable function:X1+X2+...Xn subject to the constraint X1^2+X2^2+...+Xn^2=1
 2 years ago
 2 years ago

This Question is Closed

mukushla Group TitleBest ResponseYou've already chosen the best response.2
aren't u supposed to use lagrangian Constrained Optimization ?
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
so u want to find extremums of \[f(x_1,x_2,...,x_n)=x_1+x_2+...+x_n\]subject to this constraint\[x_1^2+x_2^2+...+x_n^2=1\]
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
set up ur lagrangian\[L=f\lambda g\]and consequencly ur equations\[x_1^2+x_2^2+...+x_n^2=1\]\[\frac{\partial L}{\partial x_1}=0\]\[\frac{\partial L}{\partial x_2}=0\]...\[\frac{\partial L}{\partial x_n}=0\]these n+1 equation with n+1 unknown (degree of freedom is 0) will give u the values of \[x_1,x_2,...,x_n,\lambda\]for which \(f\) is max or min
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
still one missed point\[g(x_1,x_2,...,x_n)=x_1^2+x_2^2+...+x_n^21\] plz let me know if u got it from here
 2 years ago

mymathcourses Group TitleBest ResponseYou've already chosen the best response.1
yeah, but why there is n+1 equations?
 2 years ago

mymathcourses Group TitleBest ResponseYou've already chosen the best response.1
we also have dL/d入=x1^2+x2^2+...xn^21 right?
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
n partial derivatives and the constrained itself is one of the equations so u have n+1 equation
 2 years ago

mymathcourses Group TitleBest ResponseYou've already chosen the best response.1
and we also have dL/d入=g right? @mukushla
 2 years ago

mymathcourses Group TitleBest ResponseYou've already chosen the best response.1
then i got x1^2+x2^2+....xn^2=1
 2 years ago

mymathcourses Group TitleBest ResponseYou've already chosen the best response.1
L=f+g入
 2 years ago

mymathcourses Group TitleBest ResponseYou've already chosen the best response.1
then dL/dx1 =1+2x1入 but not zero?
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
one of equations for example\[\frac{\partial L}{\partial x_1}=0\]\[\frac{\partial f}{\partial x_1}\lambda\frac{\partial g}{\partial x_1}=0\]\[12\lambda x_1=0\]set up other equations like this anf find \(\lambda\) first
 2 years ago

mymathcourses Group TitleBest ResponseYou've already chosen the best response.1
ok, so what I did is:L=g+fλ, and then x1=x2=x3=...xn=1/(2λ) for the constraint function, we can get nX1^2=1, then x=(1/n)^0.5, then I plug this in to L again, the function L can write as L=nX1+(1/2λ)(nX1^21) after plugging into x=(1/n)^0.5 i FINALLY get L=n^0.5 so there is only one answer, but I have to find maximum and min, So I really get confused, can you give me more hint? appreciates a lot!
 2 years ago

mymathcourses Group TitleBest ResponseYou've already chosen the best response.1
maybe i made a mistake, x=(+or ) n^0.5 right, so there r 2 answer for L, n^0.5 and n^0.5? so one is max and the other one is min, am I RIGHT?
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
emm...what i know as standard form for L is fgλ
 2 years ago

mymathcourses Group TitleBest ResponseYou've already chosen the best response.1
f+gλ or fgλ doesnt matter, cause the answer is same. if gλ, then g=0, if gλ then g=0, g also =0, so this isnt the point
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
ok u r right it doesn't matter and it gives\[x_1=x_2=...=x_n=\frac{1}{2\lambda}\]and plugging this in\[x_1^2+x_2^2+...+x_n^2=1\]gives\[n\frac{1}{4\lambda^2}=1\]and\[\lambda=\pm \frac{\sqrt{n}}{2}\]so the max of \(f\) is when\[x_1=x_2=...=x_n=\frac{1}{\sqrt{n}}\]and the min is when\[x_1=x_2=...=x_n=\frac{1}{\sqrt{n}}\]
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
and note that u want to find max and min of f not L
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
\[\text{max} \ (f)=\frac{n}{\sqrt{n}}=\sqrt{n}\]and\[\text{min} \ (f)=\frac{n}{\sqrt{n}}=\sqrt{n}\]
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
i hope its clear
 2 years ago

mymathcourses Group TitleBest ResponseYou've already chosen the best response.1
@mukushla yes, we got the same answer, thx a lot!!!!
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
no problem :)
 2 years ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.