1. Lagrange Multiplier

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

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

Here's what I have so far:

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

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

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

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

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

$\displaystyle 5xy = -4xy$

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

2. Originally Posted by Macleef
Use the method of Lagrange multipliers to find the min value of:

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

Here's what I have so far:

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

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

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

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

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

$\displaystyle 5xy = -4xy$

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

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

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

and from the second

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

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

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

Now choose the one that give the min.

3. Originally Posted by Macleef
Use the method of Lagrange multipliers to find the min value of:

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

Here's what I have so far:

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

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

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

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

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

$\displaystyle 5xy = -4xy$

and now... I don't know how to isolate for x or y.
On a side note, the step (**) is ok provided that $\displaystyle x \ne 0$ so this would have to be considered as a special case.