Interesting Lagrange Problem

[Tracker]

Lagrange Problem

 

max [math] y = x_1^2 + x_2^2[/math] such that [math] \frac{x_1^2}{25} + \frac{x_2^2}{9} -1 =0 [/math]

 

[math] \frac{df}{dx_1} = 2x_1 + \frac{2x_1\lambda}{25}[/math] (1)

 

[math] \frac{df}{dx_2} = 2x_2 + \frac{2x_2\lambda}{9}[/math] (2)

 

[math] \frac{df}{d\lambda} = \frac{x_1^2}{25} + \frac{x_2^2}{9} - 1 [/math] (3)

 

Dividing equation [math] \frac{1}{2} [/math]

 

[math] \frac{x_1}{x_2} = \frac{9*x_1}{25*x_2} [/math] As you can see the x_1 and x_2 just cancels out. Do you know what the issue is? I tried using a quadratic, but it still has the lambda in it, so I cannot solve it that way. How can I solve this one using the Lagrange?

Edited September 26, 2012 by Tracker

[imatfaal]

Graphically - [math] \frac{x_1^2}{25} + \frac{x_2^2}{9} -1 =0 [/math] describes an elipse centred at (0,0) and [math] Constant = x_1^2 + x_2^2 [/math] describes a circle centred at (0,0) the maximum value the constant (ie the largest radius) can take and still be constrained by the elipse is when the circle is tangential to the ellipse at the end of the semi-major axis. But you wanted a solution using Lagrange multiplier

 

I would suggest that you put every stage up on the boards. Remember that the method you have shown so far will give the stationary points (not only maxima) so you should expect more than one. The wikipedia entry on the Lagrange multiplier might help you.

[Tracker]

imatfaal we can use the equation I wrote down to figure out x1 or x2 is equal to zero when we have the maximum. From basic understanding we know the two critical points are going to be +/-5 and +/-3, but they will both be positive because the max is squared for each variable. I just don't understand how to finish this off with lagrange. I logically can figure out the answer.

[Tracker]

Lol I found the mistake. It was a mistake on the paper when I wrote it down, but I posted the right equation.....

[imatfaal]

Excellent!

 

I am no mathematician - but when things inexplicably refuse to make sense it is is always useful to revisit the basic items; R-the-Q (read the question), Signs! (check your signs), Line-by-Line (is the transcription of the boring part correct from line to line) etc...