LaGrange Multiplier Question HELP! <- What am I doing wrong?

Please help on where I am going wrong.

Question is:

$T = x^2 +sin^2\theta -z, z = 2x^2sin^2\theta + 1$


Use Lagrange Multipliers to find out its critical points.

My working out.

This formed

$L(x,\theta,\lambda) = x^2 +sin^2\theta -\lambda(2x^2sin^2\theta + 1)[br][br]L_x = 2x - \lambda(4xsin^2\theta)[br]L_\theta = 2cos\theta sin\theta - \lambda(4x^2cos\theta sin\theta)[br]L_\lambda = -2x^2sin^2\theta - 1[br][br]Critical when L_x= L_\theta = L_\lambda = 0$

First Equation:

$2x - \lambda(4xsin^2\theta) = 0[br][br]2x(1-2\lambda sin^2\theta) = 0[br][br]x = 0 or 1 - 2\lambda sin^2\theta = 0 [br][br]so x=0 and sin^2\theta = \frac{1}{2\lambda}$

Second Equation:

$L_\theta = 2cos\theta sin\theta - \lambda(4x^2cos\theta sin\theta) = 0[br][br]Divide by 2cos\theta sin\theta[br][br]Gives \lambda = \frac{1}{2x^2}$

Final Equation:

$L_\lambda = -2x^2sin^2\theta - 1 = 0[br][br]Rearranging Gives: sin^2\theta = \frac{-1}{2x^2}$

Now this doesn't solve for some reason.

Could not upload the question for some reason due to server connection problem.

Thanks

T is a function of x, z and theta

Lamda is a parameter... You are differentiating with respect to lamda

So my L_\lambda

well I am trying to follow your workings because there several different ways you can set your work

If you work a Lagrangian L, the last equation it merely gives your constraint (trivial to even bother to write it.


In this problem there should be 3 equations from partial differentiation
diff wrt x
diff wrt z
diff wrt theta

PLUS the constraint equation

4 equations and 4 unknowns, x, z, theta and lamda

Is there a range in the value of theta in the problem?

nope no range given so I guess the first value....

so there should also be L_z = -1

the partial differentiation wrt to z simply yields lamda = -1

then the rest is trivial so long as you have a sensible range for the values of theta

so is this correct?

$[br][br]T_x = 2x[br][br]T_\theta = 2sin\theta cos\theta[br][br]T_z = -1[br][br]T_\lambda = -2x^2sin^2\theta - 1$

this is my partial solution ...
I hope there are no errors

EDIT It will not let me upload!!! It is your unlucky day..

well.....any other ways? or maybe upload whenever the server is fixed

There is something wrong with TSR uploading images ...

I wrote a partial solution about 2/3 of the way but I I cannot upload until then

Alternative is to email me in my website's email and I will attach back the solution

There is something wrong with TSR uploading images ...

I wrote a partial solution about 2/3 of the way but I I cannot upload until then

Alternative is to email me in my website's email and I will attach back the solution

There is something wrong with TSR uploading images ...

I wrote a partial solution about 2/3 of the way but I I cannot upload until then

Alternative is to email me in my website's email and I will attach back the solution

Oh I see what you meant.

For some reason, I don't know why I did not recognize this.