Lagrange multipliers, guidance needed

[ilyas.h]

Homework Statement

f(x,y) is function who's mixed 2nd order PDE's are equal.

consider k_f:

determine the points on the graph of the parabloid f(x,y) = x^2 + y^2 above the ellipse 3x^2 + 2y^2 = 1 at which k_f is maximised and minimised.

The Attempt at a Solution

is this the langrange equation that I'd use?

L(x,y,\lambda )=k_{f}(x,y)-\lambda k_{g}(x,y)

or maybe it's just:
L(x,y,\lambda )=k_{f}(x,y)-\lambda g(x,y)

im not sure if I should put the g function (3x^2 + 2y^2 = 1) into the k_f. If I do need to do this, then the algebra is very ugly.

 

[Ray Vickson]

Homework Statement

f(x,y) is function who's mixed 2nd order PDE's are equal.

consider k_f:

determine the points on the graph of the parabloid f(x,y) = x^2 + y^2 above the ellipse 3x^2 + 2y^2 = 1 at which k_f is maximised and minimised.

The Attempt at a Solution

is this the langrange equation that I'd use?

L(x,y,\lambda )=k_{f}(x,y)-\lambda k_{g}(x,y)

or maybe it's just:
L(x,y,\lambda )=k_{f}(x,y)-\lambda g(x,y)

im not sure if I should put the g function (3x^2 + 2y^2 = 1) into the k_f. If I do need to do this, then the algebra is very ugly.

Have you made an attempt to simplify $k_f(x,y)$ as much as possible before using it in a Lagrangian or anywhere else? Far from ugly, the algebra will be very simple and straightforward.

 

[ilyas.h]

Have you made an attempt to simplify $k_f(x,y)$ as much as possible before using it in a Lagrangian or anywhere else? Far from ugly, the algebra will be very simple and straightforward.

my k_f(x,y) is:

\frac{4}{(1+4x^{2}+4y^{2})^2}

How would you simplify this/the problem? I tried simplifying with wolfram alpha, it doesn't return anything useful.

my k_g(x,y) is:

\frac{24}{(1+36x^{2}+16y^{2})^2}

please, someone help. It's a bit of an emergency.

 

[Ray Vickson]

my k_f(x,y) is:

\frac{4}{(1+4x^{2}+4y^{2})^2}

How would you simplify this/the problem? I tried simplifying with wolfram alpha, it doesn't return anything useful.

my k_g(x,y) is:

\frac{24}{(1+36x^{2}+16y^{2})^2}

please, someone help. It's a bit of an emergency.

Why do you care about $k_g$? It has nothing at all to do with the question as originally posed. Also: knowing the form of $k_f$ should allow you to simplify the problem significantly, to get a problem that is much easier than the original one. I cannot say more, without essentially doing the problem for you

 

[ilyas.h]

Why do you care about $k_g$? It has nothing at all to do with the question as originally posed. Also: knowing the form of $k_f$ should allow you to simplify the problem significantly, to get a problem that is much easier than the original one. I cannot say more, without essentially doing the problem for you

ok, then the Lagrange equation we are interested in is:

L(x,y, \lambda)=\frac{4}{(1+4x^{2}+4y^{2})^2}- \lambda(3x^{2} + 2y^{2}-1)
what do you mean by form of k_f(x,y)? could you come up with a mini-alternative example to what you mean, I just don't see how you can simplify it, unless this has something unique to do with the problem as opposed to general algebraic manipulation.

 

[Ray Vickson]

ok, then the Lagrange equation we are interested in is:

L(x,y, \lambda)=\frac{4}{(1+4x^{2}+4y^{2})^2}- \lambda(3x^{2} + 2y^{2}-1)
what do you mean by form of k_f(x,y)? could you come up with a mini-alternative example to what you mean, I just don't see how you can simplify it, unless this has something unique to do with the problem as opposed to general algebraic manipulation.

OK, so don't simplify it; just solve it as-is. It does work, but takes a bit more effort.

 

[ilyas.h]

OK, so don't simplify it; just solve it as-is. It does work, but takes a bit more effort.

\frac{\partial L}{\partial x}= (-64x(1+4x^{2}+4y^{2})^{-3}) - 6x\lambda = 0

\frac{\partial L}{\partial y}= (-64y(1+4x^{2}+4y^{2})^{-3}) - 4y\lambda = 0

\frac{\partial L}{\partial \lambda}= 1 - 3x^{2}-2y^{2}=0

this leads to a contradiction though, using the first and second equation we get:

(\frac{-32}{3}(1+4x^{2}+4y^{2})^{-3}) = \lambda = ((-24)(1+4x^{2}+4y^{2})^{-3})

 

[Ray Vickson]

\frac{\partial L}{\partial x}= (-64x(1+4x^{2}+4y^{2})^{-3}) - 6x\lambda = 0

\frac{\partial L}{\partial y}= (-64y(1+4x^{2}+4y^{2})^{-3}) - 4y\lambda = 0

\frac{\partial L}{\partial \lambda}= 1 - 3x^{2}-2y^{2}=0

this leads to a contradiction though, using the first and second equation we get:

(\frac{-32}{3}(1+4x^{2}+4y^{2})^{-3}) = \lambda = ((-24)(1+4x^{2}+4y^{2})^{-3})

No contradiction; you have just made some fatal errors.

You can write
0= L_x = -Rx - 6x \lambda , \; \text{and} \; 0 = L_y = -Ry - 4 y \lambda
where $R = 64/(1 + 4x^2+4 y^2)^3$. You have attempted to convert $-Rx - 6 x \lambda = 0$ into $-R - 6 \lambda = 0$ (similarly for $y$), and you are NOT allowed to do that without some extra conditions.

 

[ilyas.h]

1. \frac{\partial L}{\partial x}= (-64x(1+4x^{2}+4y^{2})^{-3}) - 6x\lambda = 0

2. \frac{\partial L}{\partial y}= (-64y(1+4x^{2}+4y^{2})^{-3}) - 4y\lambda = 0

3. \frac{\partial L}{\partial \lambda}= 1 - 3x^{2}-2y^{2}=0

if you get an equation for lambda from the first and second equation without dividing through by any variables you get that (using \lambda):

\frac{32}{3}xy(1+4x^{2}+4y^{2})^{3} = -24xy(1+4x^{2}+4y^{2})^{3}

how am I meant to get a relationship between x and y from the above equation that I can utilize in the third equation?

 

[Ray Vickson]

1. \frac{\partial L}{\partial x}= (-64x(1+4x^{2}+4y^{2})^{-3}) - 6x\lambda = 0

2. \frac{\partial L}{\partial y}= (-64y(1+4x^{2}+4y^{2})^{-3}) - 4y\lambda = 0

3. \frac{\partial L}{\partial \lambda}= 1 - 3x^{2}-2y^{2}=0

if you get an equation for lambda from the first and second equation without dividing through by any variables you get that (using \lambda):

\frac{32}{3}xy(1+4x^{2}+4y^{2})^{3} = -24xy(1+4x^{2}+4y^{2})^{3}

how am I meant to get a relationship between x and y from the above equation that I can utilize in the third equation?

For any real $x,y$ we have $1 + 4 x^2 + 4 y^2 \geq 1 > 0$, so you are allowed to divide it out on both sides, to end up with
\frac{32}{3} xy = - 24 xy
What does that tell you?

 

[ilyas.h]

For any real $x,y$ we have $1 + 4 x^2 + 4 y^2 \geq 1 > 0$, so you are allowed to divide it out on both sides, to end up with
\frac{32}{3} xy = - 24 xy
What does that tell you?

that x =y = 0, however this cannot be true due to the third equation.

therefore x and y has to take multiple values that satisfy the following relationships:

\frac{32}{3}xy = -24xy

1 - 3x^{2}-2y^{2}=0

going along the right lines?

and thanks for the help so far.

 

[Ray Vickson]

that x =y = 0, however this cannot be true due to the third equation.

****************************************************************************
That is not what it tells me. Back to the drawing board.

*****************************************************************************

therefore x and y has to take multiple values that satisfy the following relationships:

\frac{32}{3}xy = -24xy

1 - 3x^{2}-2y^{2}=0

going along the right lines?

and thanks for the help so far.

 

[ilyas.h]

...

ok, so then x must equal y which must equal zero.

Or perhaps my calculations are wrong...

 

[ilyas.h]

...

 

[ilyas.h]

someone please help me!