Find the smallest and highest value of the product $xyz$ Find the smallest and highest value of the product $xyz$ assuming that:
$x + y + z = 10$ and
$x^2 + y^2 + z^2 = 36$.
I calculated this:
$x+y+z=10 => (x+y+z)^2=10^2$
$x^2+y^2+z^2+2xy+2yz+2zx=100$
$(x^2+y^2+z^2+2xy+2yz+2zx)-(x^2+y^2+z^2)=100-36$
$2xy+2yz+2zx=64$
$xy+yz+zx=32$
I'm stuck. What is the next step to this exercise?
My idea is to show the equation using one variable and after computing the derivative reach global extremes.

It seems like a better start if you eliminate a variable (or two...)
Thus $z= 10-x-y$, so $x^2+y^2+100-2(x+y)+(x+y)^2=36$, that is, $x^2+y^2+xy-x-y+64=0$.
You are looking for the maximum of $xy(10-x-y)$. 
If you do not want to look for anything creative, Lagrange multiplier surely works. 
