<s>[INST] 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.
 [/INST] We have $$x^2+y^2=36-z^2$$ and $$x+y=10-z,$$ which gives
$$(10-z)^2-2xy=36-z^2$$ or
$$xy=32-10z+z^2$$ and $$xyz=32z-10z^2+z^3.$$
Also, $$(x+y)^2\geq4xy,$$ which gives
$$3z^2-20z+28\leq0$$ or
$$2\leq z\leq\frac{14}{3}.$$
Can you end it now?
I got $\max(xyz)=\frac{896}{27}$ and $\min(xyz)=32$.
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