# Math Help - partial differentials

1. ## partial differentials

Hey folks,

I'm working on this problem from the partial diff section of my book:

Verify that the sum of three numbers, x, y, and z, whose product is constant, is a minimum when they are all equal.

Just need a hint, I hope, as what I've tried, I'm not sure makes sense.

I figure I need an equation that somehow includes the constant product restriction in the sum. Maybe something like:

$u = x + y + z$
$k = xyz$
and then,
$u = \frac{k}{yz} + \frac{k}{xz} + \frac{k}{xy}$

Then find the total differential, and so on. I know when finding a min/max, each partial has to equal zero. But this one's getting me a little confused.

Scott

2. Originally Posted by ScottO
$u = x + y + z$
$k = xyz$
and then,
$u = \frac{k}{yz} + \frac{k}{xz} + \frac{k}{xy}$
Notice that,
$x+y+z \geq 3\sqrt[3]{xyz} =3\sqrt[3]{k}$
So that is the minimum value.