having some trouble with this, but I dont know what else goes in the equation besides , 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) )