Question

Lagrange Multiplier
Million Dollar question.
Anybody please
Use the method of Lagrange multipliers to find the max and min values
of g(s,t)=t^2e^s given that s^2+t^2=3 . How are we assured that these
extreme values exist?

Answer 1

I have one critical point

Answer 2

Are you looking for some real analysis kindof existence proof.. or just some intuition ?

Answer 3

real Analysis please

Answer 4

Then what is lagrange multiplier doing here ?

Answer 5

\[(1, \sqrt{2}) (1 -\sqrt{2})\]

Answer 6

Pardon?

Answer 7

Lagrange multiplier method is just one way to find the extrema.

It has nothing to do with the existence part right ?

Answer 8

yes I know, and from the Extreme Value Theorem tells us that minimums and maximums will always exist.

Answer 9

I think my Instructor wants me to graph, and show that the max and min exist.

Answer 10

But I am only getting max and no mean from the two critical point \[(1 \sqrt{2}) (1 \sqrt{-2})\] and f gives me 2e for both points

Answer 11

Is this max or min point? How can I tell.

Answer 12

tried to graph but looks tedious.

Answer 13

and s also =-3 which has not value for t.

Answer 14

hello.

Answer 15

Lagrange multiplier doesn't give you information about the type of extrema. You will need to use hessian matrix for that

Answer 16

Hessian? did not cover that

Answer 17

Then you will need to find the type of extrema by some other means

Answer 18

but
To determine if we have maximums or minimums we just need to plug these into the function. critical points

Answer 19

Yes that will do

Answer 20

yes that that is what I did.

Answer 21

Can you determine the max and min please?

Answer 22

I got 2e for boths critical points

Answer 23

Are the two points bot max? what about min. could it be that these function doest not have a min.

Answer 24

I am getting 6 critical points
http://www.wolframalpha.com/input/?i=solve+t%5E2e%5Es%3D2%5Clambda*s%2C+2te%5Es+%3D+2%5Clambda*t%2C+s%5E2%2Bt%5E2%3D3

Answer 25

please look closely at x=-3 the value for t -root

Answer 26

\((\pm \sqrt{3}, 0 )\) give you minimum

\((1,\pm \sqrt{2})\) give you maximum

and yeah other two critical points are not real

Answer 27

I figured that simply by evaluating g(s,t) at those values, as you said earlier

Answer 28

And the last part of the question how are we sure that these values exist?

Answer 29

How did you get the second critical point pmsruareroont of 3.

Answer 30

Please go through this for a proof of lagrange multipliers method
http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-c-lagrange-multipliers-and-constrained-differentials/session-40-proof-of-lagrange-multipliers/MIT18_02SC_notes_22.pdf

Answer 31

I got those critical points by solving :

http://www.wolframalpha.com/input/?i=solve+t%5E2e%5Es%3D2%5Clambda*s%2C+2te%5Es+%3D+2%5Clambda*t%2C+s%5E2%2Bt%5E2%3D3

Answer 32

\(\sqrt{3} = 1.73205\ldots\)

Answer 33

geometrical look at this problem is helpful. z= f (x, y)= x + 2y is a plane
passing through the origin. The constraint x^2 +y ^2 + 5 is a right circular
cylinder. The maximum value of f (x, y)= x+ 2y exists and occurs at the
highest point on the curve of intersection of the cylinder and the plane. Example from my note. I do not need proof of lagrange to know if the values exist or not. i need like a geometrical look. @ganeshie8 and thank you so much. really apprecaite.

Answer 34

geometric look?

Answer 35

http://www.wolframalpha.com/input/?i=graph+t%5E2e%5Es

form here I can see the two values of the max and min

How can I plot this on paper. @ganeshie8 thank you.

Answer 36

@mathmale can you please look at this lagrange multiplier example. I get everything except the last part of the question. "How are we sure these extreme values exist" I know by the extreme value theorem a max and min WILL ALWAYS EXIST.

Thank you

Answer 37

What will be the argument here? I could not sketch of the graph of function but from wolfram I can see that the function has two max value and one minimum value.

http://www.wolframalpha.com/input/?i=graph+t%5E2e%5Es @mathmale

Thanks

Answer 38

@mathmale help please.

Answer 39

I'd love to help, but it's been several years since I last encountered Lagrange Multipliers. Frankly, I don't remember enough about them to help you. Thanks for calling on me, however. Good luck!

Answer 40

Its Ok

Thanks