<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] Consider the equation: $A^3-10A^2+32A-xyz=0$, whose roots are $x,y,z$. We need to find the maximum and minimum values of:
$$xyz=A^3-10A^2+32A=f(A).$$ 
FOC:
$$f'(A)=3A^2-20A+32=0 \Rightarrow A_1=\frac83; \ \ A_2=4.$$
SOC:
$$\begin{align}f''(A)&=6A-20; \\
f''\left(\frac83\right)&=-4<0 \Rightarrow f\left(\frac83\right)=\frac{896}{27} \ (\text{max});\\
f''(4)&=4>0 \Rightarrow f(4)=32 \ (\text{min}).\end{align}
$$
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