method of Lagrange multipliers question

Printable View

May 14th 2010, 06:38 PM
quah13579

method of Lagrange multipliers question

Can someone tell me how to use method of Lagrange multipliers to do following question:

Use the method of Lagrange multipliers to find the critical points of the following constrained optimisation problems. Do not attempt to test the nature of these critical points.

(a) Optimise f(x, y) = x + y,
subject to the constraint: x^2 + y^2 - 4 = 0

(b) Optimise f(x, y, z) = x^2 + y^2 + z^2,
subject to the constraints: x + y + z = 1 and x + 2y + 3z = 2
May 14th 2010, 09:06 PM
simplependulum

Quote:

Originally Posted by quah13579

Can someone tell me how to use method of Lagrange multipliers to do following question:

Use the method of Lagrange multipliers to find the critical points of the following constrained optimisation problems. Do not attempt to test the nature of these critical points.

(a) Optimise f(x, y) = x + y,
subject to the constraint: x^2 + y^2 - 4 = 0

(b) Optimise f(x, y, z) = x^2 + y^2 + z^2,
subject to the constraints: x + y + z = 1 and x + 2y + 3z = 2

(a)

$f = x+y + a ( x^2 + y^2 - 4 )$

we have

$f_x = 1 + 2ax = 0$ and

$f_y = 1 + 2ay = 0$

$x = y = - \frac{1}{2a}$

Sub. $x=y$ into the constraint $x^2 + y^2 = 4$

$2x^2 = 4 , x = \sqrt{2} ~ or~ -\sqrt{2}$

$(x,y) = ( \sqrt{2}, \sqrt{2}) ,( -\sqrt{2}, \sqrt{2}) ,( \sqrt{2}, -\sqrt{2}),( -\sqrt{2}, -\sqrt{2})$

(b)

$f= x^2 + y^2 + z^2 + a( x + y + z -1 ) + b( x + 2y + 3z -2)$

$f_x = 2x + a + b = 0$ (1)
$f_y = 2y + a + 2b = 0$ (2)
$f_x = 2z + a + 3b = 0$ (3)

$x = -\frac{1}{2} (a+b)$
$y = -\frac{1}{2} (a+2b)$
$z = -\frac{1}{2} (a+3b)$

It contructs a plane , also from the two constraints , we have three planes now , solve the system of three linear equations .

$(x,y,z) = ( \frac{1}{3} , \frac{1}{3} , \frac{1}{3} )$
May 14th 2010, 09:35 PM
11rdc11

Quote:

Originally Posted by quah13579

Can someone tell me how to use method of Lagrange multipliers to do following question:

Use the method of Lagrange multipliers to find the critical points of the following constrained optimisation problems. Do not attempt to test the nature of these critical points.

(a) Optimise f(x, y) = x + y,
subject to the constraint: x^2 + y^2 - 4 = 0

(b) Optimise f(x, y, z) = x^2 + y^2 + z^2,
subject to the constraints: x + y + z = 1 and x + 2y + 3z = 2

A)

$1 = \lambda 2x$

$1 = \lambda 2y$

$x^2 + y^2 = 4$

$x = \frac{1}{2\lambda} = y$

so plug into your other equation

$x^2 + x^2 =4$

$2x^2 =4$

$x^2 = 2$

$x = ^+_- \sqrt{2}$

so your points are

$(\sqrt{2},\sqrt{2}),(-\sqrt{2},\sqrt{2}),(-\sqrt{2},-\sqrt{2}),(\sqrt{2},-\sqrt{2})$

Now just see which is you maxs and mins.

Try following my approach for the next problem.