To find the maximum and minimum value of x such that it satisfies a polynomial Find the maximum and minimum value of $x$, where:


*

*$x+y+z=4$

*$x^2+y^2+z^2 =6$


I thought I could use these values to form a equation having $x,y,z$ as roots and the sum of roots and $\sum{xy}$ but could not get any idea about the product of roots hence could not move ahead.

Let we set $x=\frac{4}{3}+u,y=\frac{4}{3}+v,z=\frac{4}{3}+w$. The given constraints translate into:
$$\left\{\begin{array}{rcl}v+w &=& -u \\ v^2+w^2 &=& \frac{2}{3}-u^2\end{array}\right.$$
or:
$$\left\{\begin{array}{rcl}v+w &=& -u \\ v\cdot w &=& u^2-\frac{1}{3},\end{array}\right.$$
so $v,w$ are roots of the polynomial $p(t)=t^2+ut+\left(u^2-\frac{1}{3}\right)$. They are real roots provided that $u^2-4\left(u^2-\frac{1}{3}\right)\geq 0$, so the critical values for $u$ are given by $u=\pm\frac{2}{3}$ and:
$$ x\in\left[\frac{2}{3},2\right].$$
