# Math Help - practical differential problem

1. ## practical differential problem

having some trouble with this, but I dont know what else goes in the equation besides $XY=-12$, can someone help?

"Find two numbers whose product is -12 and the sum of whose square is a minimum."

i mostly dont know what to do on the last part

2. Originally Posted by Arez
having some trouble with this, but I dont know what else goes in the equation besides $XY=-12$, can someone help?

"Find two numbers whose product is -12 and the sum of whose square is a minimum."

i mostly dont know what to do on the last part
Are you tying to mininize $x^2+y^2$ given $XY=-12$?

3. Originally Posted by Arez
having some trouble with this, but I dont know what else goes in the equation besides $XY=-12$, can someone help?

"Find two numbers whose product is -12 and the sum of whose square is a minimum."

i mostly dont know what to do on the last part

xy= -12 => y=-12/x
x^2 + y^2 =z
x^2 + (-12/x)^2=z
differentiate z with respect to x.
dz/dx = 2x - 288/(x^3)
Now consider minimization criteria,
z'=0 and z''>0 (z'=dz/dx and z'' is second derivative).
Hence,
For z'=0 we get,
x^4 = 144
Solve this for x.

Now z''=2 + 864/(x^4)
This is always positive, so lets take x= + sq.root(12),
then y= - sq.root(12), or
(x, y)= (+ sq.root(12), - sq.root(12) ).

The other three solutions for (x,y) are,

(- sq.root(12), + sq.root(12) )
(+ sq.root(-12), - sq.root(-12) )
(- sq.root(-12), + sq.root(-12) )

4. I'm still not sure what the orginal question was.

5. Originally Posted by matheagle
I'm still not sure what the orginal question was.
I think it meant find two numbers whose product is -12 and the sum of those two numbers is minimum.

6. Originally Posted by matheagle
I'm still not sure what the orginal question was.
I suspect it's (x + y)^2 that is to be minimised ....

7. Originally Posted by niranjan
I think it meant find two numbers whose product is -12 and the sum of those two numbers is minimum.
I meant sum of the squares of those numbers to be minimum i.e.,
$x^2 +y ^2$

8. Originally Posted by akshatha
I meant sum of the squares of those numbers to be minimum i.e.,
$x^2 +y ^2$

I figured that

9. Originally Posted by akshatha
I meant sum of the squares of those numbers to be minimum i.e.,
$x^2 +y ^2$
Yes, I mean the same. I quoted wrongly that its sum of numbers, but its sum of squares of numbers which has to be minimized. I did math work for squares, but misquoted it.