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alffer1
Group Title
Multivariable Help:
Once again I'm stumbling on Lagrange Multipliers. I need to find the maximum and minimum distance from the origin of a point on the ellipsoid 4x^2 + (y2)^2 + (z1)^2 = 4.
I've determined that the partials are as follows:
df/dx = 2x
df/dy = 2y
df/dz = 2z
dg/dx = 8x
dg/dy = 2(y2)
dg/dz = 2(z1)
From this, I get the Lagrange equations to be:
2x = L(8x)
2y = L(2y4)
2z = L(2z2)
From these equations, I get x = 0, L = 1/4, y = 2/3 and z = 1/3. Where do I go from here?
 11 months ago
 11 months ago
alffer1 Group Title
Multivariable Help: Once again I'm stumbling on Lagrange Multipliers. I need to find the maximum and minimum distance from the origin of a point on the ellipsoid 4x^2 + (y2)^2 + (z1)^2 = 4. I've determined that the partials are as follows: df/dx = 2x df/dy = 2y df/dz = 2z dg/dx = 8x dg/dy = 2(y2) dg/dz = 2(z1) From this, I get the Lagrange equations to be: 2x = L(8x) 2y = L(2y4) 2z = L(2z2) From these equations, I get x = 0, L = 1/4, y = 2/3 and z = 1/3. Where do I go from here?
 11 months ago
 11 months ago

This Question is Closed

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
I'm a little rusty, but it looks like you found the closest point to the origin that is on the ellipsoid. So follow Pythagorus\[d=\sqrt{x ^{2}+y ^{2}+z ^{2}}\]
 11 months ago

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
Oh my... *Pythagoras
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
ok for that I get sqrt(5)/3. Now how do I get the farthest point?
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
you didn't get the critical points yet it seems. actually I think they will be boundaries... let's see...
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
ok now I got myself totally confused...
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
yeah you did confuse things a tad... constraint:\[f(x)=4x^2+(y2)^2+(z1)^2=4\]to optimize we'll use distance squared\[g(x)=x^2+y^2+z^2\]we then apply Lagrange's trick to get\[\\\begin{align}&\nabla f(x)&=\lambda\nabla g(x)\\(1)&8x&=2\lambda x\\(2)&2y4&=2\lambda y\\(3)&2z4&=2\lambda z\\(4)&4x^2+(y2)^2+(z1)&=4\end{align}\]
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
wait...I thought the Lambda was put on the side of the constraint derivatives...
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
yep, my bad, let me fix that :P
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
ok the heck with using the latex, takes too long you had it right: 2x = L(8x) 2y = L(2y4) 2z = L(2z2) 4x^2+(y2)^2+(z1)^2=4 it's a matter of solving this system: 4 equations, 4 unknowns
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
and I solved that and got the 4 values shown above...
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
so what do those 4 values mean right now?
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
they are a set of critical points, but I'm not sure they are the only ones...
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
well, just one critical point I mean
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
lambda means nothing as I mentioned before
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
But if there is only one critical point, how can there be a max distance AND a min distance?
 11 months ago

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
I have this tiny little voice in my head that keeps saying, "go through the constraint". I know it sounds weird, but I remember something about how there is always another one through the constraint.
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
well if you plug x=0 into the restraint you get a circle...
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
ohhhhhhhh I think I see the problem again it is a matter of the ambiguity in the sign...
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
how did you solve for y ?
 11 months ago

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
haha... plugged in lambda=1/4
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
um...I plugged in L = 1/4, got 2y = (y/2) 1
 11 months ago

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
1/4 would work for the x equation too, huh?
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
Where on earth did the negative come from?
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
oh I see what you mean
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
since x is 0, the sign of Lambda doesn't matter...
 11 months ago

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
maybe/ maybe not... but I think that when you divide out the x, you eliminate solutions.
 11 months ago

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
Not to confuse the discussion, but I think there are two maximums.
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
see, you need to plug in that x value into the restraint, using that value of lambda 2x = L(8x) > L=1/4, x=0 2y = L(2y4) > y2=y/L 2z = L(2z2) > z1=z/L plug these into 4x^2+(y2)^2+(z1)^2=4 now look how we can rewrite the restraint:
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
4=(y/L)^2+(z/L)^2
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
circle of radius 2 ?
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
well we need to deal with L first
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
L=1/4 so 64=y^2+z^2=8^2
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
now we need a way to represent y in terms of z or vice versa
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
or not... but that's the plan
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
ok I'm working this out...
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
me too :P
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
yeah that works\[\lambda={y\over y1}={z\over z1}\]
 11 months ago

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
I think that might be y2 instead of y1
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
oh yes, sorry!
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
also it should be y^2+z^2=1/4 after multiplying by L^2 this is why we always need extra eyes on these things!
 11 months ago

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
I was just looking at that... thank goodness you saw it too.
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
wait, now are we supposed to solve for y and z?
 11 months ago

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
yeah, this is crazy. I've filled pages with mostly garbage.
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
oh y = 2z haha
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
that's what i get, so plugging that into the restraint gives you...?
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
z = sqrt(20)/20 y = sqrt(20)/10
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
those are both positive negative by the way
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
is that right?
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
yeah that's what I get, though I am disturbed that wolfram disagrees
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
oh boy...
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
oh I see, we should have just plugged this directly into the original restraint with x=0 4=(y2)^2+(z1)^2
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
oh oh ok that makes more sense
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
went a little overboard trying to simplify I guess
 11 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.1
ok gotta go. Thanks so much!
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
with the +/ you have your max and min of course welcome!
 11 months ago

EulerGroupie Group TitleBest ResponseYou've already chosen the best response.1
Thanks for brushing out some cobwebs... more to go.
 11 months ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.2
ikr? seems like history...
 11 months ago
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