## SUBJECT 2

Let $\mathrm{x} \in \mathrm{R}$ with $|\mathrm{x}-1| \leq \frac{1}{5}$. Determine $x$ if $\sqrt[3]{(6-5 x)(4 x-3)}+\sqrt[3]{(5 x-4)(5-4 x)}=2$. Prof. Alexandru Cărnaru
## SUBJECT 2

From $|\mathrm{x}-1| \leq \frac{1}{5}$ or $x \in\left[\frac{4}{5}, \frac{6}{5}\right]$ we obtain that all parentheses are positive

and we apply the inequality of means

$\sqrt[3]{(6-5 x)(4 x-3) \cdot 1}+\sqrt[3]{(5 x-4)(5-4 x) \cdot 1} \leq \frac{6-5 x+4 x-3+1}{3}+\frac{5 x-4+5-4 x+1}{3}=2$ $3 p$

with equality when all parentheses are equal to 1, we get $x=1$ as the unique solution

$.2 p$