Lagrange Multiplier

**Macleef** · March 29th 2009, 04:21 AM

Use the method of Lagrange multipliers to find the min value of:

$f(x, y) = 5x^2 - 4x^2$ subject to the constraint $g(x, y) = x^2 + y^2 = 9$

Here's what I have so far:

$f_x = 10x$
$f_y = -8y$
$g_x = 2x$
$g_y = 2y$

$10x = 2 \lambda x$
$-8y = 2 \lambda y$
$9 = x^2 + y^2$

$\frac{5x}{x} = \lambda$

$-\frac{4y}{y} = \lambda$

$\frac{5x}{x} = -\frac{4y}{y}$

$5xy = -4xy$

and now... I don't know how to isolate for x or y.

**Danny** · March 29th 2009, 06:35 AM

Originally Posted by Macleef

Use the method of Lagrange multipliers to find the min value of:

$f(x, y) = 5x^2 - 4x^2$ subject to the constraint $g(x, y) = x^2 + y^2 = 9$

Here's what I have so far:

$f_x = 10x$
$f_y = -8y$
$g_x = 2x$
$g_y = 2y$

$10x = 2 \lambda x$ (**)
$-8y = 2 \lambda y$
$9 = x^2 + y^2$

$\frac{5x}{x} = \lambda$

$-\frac{4y}{y} = \lambda$

$\frac{5x}{x} = -\frac{4y}{y}$

$5xy = -4xy$

and now... I don't know how to isolate for x or y.

I'll pick it up at this point (**). From these you have

$2x(\lambda - 5) = 0,\;$ $2y(\lambda+4) = 0$ subject to $9 = x^2 + y^2$ .

From the first you have $x = 0\; \text{or}\, \lambda= 5$

and from the second

$y = 0\; \text{or}\, \lambda= -4$ noting that $x = 0, y = 0$ cannot both happen due to the constraint.

If $x = 0\;\; \text{then}\; y= \pm 3$ (from the constaint)

or $y = 0\;\; \text{then}\; x= \pm 3$ (from the constaint)

Now choose the one that give the min.

**Danny** · March 29th 2009, 06:37 AM

Originally Posted by Macleef

Use the method of Lagrange multipliers to find the min value of:

$f(x, y) = 5x^2 - 4x^2$ subject to the constraint $g(x, y) = x^2 + y^2 = 9$

Here's what I have so far:

$f_x = 10x$
$f_y = -8y$
$g_x = 2x$
$g_y = 2y$

$10x = 2 \lambda x$
$-8y = 2 \lambda y$
$9 = x^2 + y^2$

$\frac{5x}{x} = \lambda$ (**)

$-\frac{4y}{y} = \lambda$

$\frac{5x}{x} = -\frac{4y}{y}$

$5xy = -4xy$

and now... I don't know how to isolate for x or y.

On a side note, the step (**) is ok provided that $x \ne 0$ so this would have to be considered as a special case.

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