# method of Lagrange multipliers question

• 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)

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

we have

$\displaystyle f_x = 1 + 2ax = 0$ and

$\displaystyle f_y = 1 + 2ay = 0$

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

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

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

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

(b)

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

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

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

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

$\displaystyle (x,y,z) = ( \frac{1}{3} , \frac{1}{3} , \frac{1}{3} )$
• 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)

$\displaystyle 1 = \lambda 2x$

$\displaystyle 1 = \lambda 2y$

$\displaystyle x^2 + y^2 = 4$

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

so plug into your other equation

$\displaystyle x^2 + x^2 =4$

$\displaystyle 2x^2 =4$

$\displaystyle x^2 = 2$

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

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