<s>[INST] 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.
 [/INST] My Solution:: Given $$x+y+z=4$$ and $$x^2+y^2+z^2=6$$
So we get $$y+z=(4-x)$$ and $$y^2+z^2 = (6-x^2)$$
Now using the Cauchy-Schwarz inequality, we get
$$\displaystyle \left(y^2+z^2\right)\cdot (1^2+1^2)\geq (y+z)^2$$
So we get $$\displaystyle (6-x^2)\cdot 2 \geq (4-x)^2$$
$$\Rightarrow\displaystyle 12-2x^2\geq 16+x^2-8x$$
$$\Rightarrow\displaystyle 3x^2-8x+4\leq 0\Rightarrow 3x^2-6x-2x+4\leq 0$$
$$\Rightarrow\displaystyle 3x(x-2)-2(x-2)\leq 0\Rightarrow x\in \left[\frac{2}{3}\;,2\right]$$
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