Let $m$ be the minimum possible value of $\log _{3}$ $\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)$, w

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Let $m$ be the minimum possible value of $\log _{3}$ $\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)$, where $y_{1}, y_{2}, y_{3}$ are real numbers for which $y_{1}+y_{2}+y_{3}=9$. Let $M$ be the maximum possible value of $\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)$, where $x_{1}, x_{2}, x_{3}$ are positive real numbers for which $x_{1}+x_{2}+x_{3}=9$. Then the value of $\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)$ is

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(8) : Using A.M. $\geq$ G.M. inequality, we have $$ \begin{aligned} & \frac{3^{y_{1}}+3^{y_{2}}+3^{y_{3}}}{3} \geq \sqrt[3]{3^{y_{1}} \cdot 3^{y_{2}} \cdot 3^{y_{3}}} \ \Rightarrow \quad & 3^{y_{1}}+3^{y_{2}}+3^{y_{3}} \geq 3 \cdot \sqrt[3]{3^{y_{1}+y_{2}+y_{3}}} \ & =3 \cdot \sqrt[3]{3^{9}}=3 \cdot 3^{3}=3^{4} \ \Rightarrow \quad & \log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}ight) \geq 4 \ 	herefore \quad & m=4 \end{aligned} $$ Again using A.M. $\geq$ G.M., we have\ $\frac{x_{1}+x_{2}+x_{3}}{3} \geq \sqrt[3]{x_{1} x_{2} x_{3}}$\ $\Rightarrow \quad x_{1} x_{2} x_{3} \leq 27=3^{3}$\ $\Rightarrow \quad \log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3} \leq 3$\ $	herefore \quad M=3$\ Now, $\log _{2}\left(m^{3}ight)+\log _{3}\left(M^{2}ight)$\ $=3 \log _{2} m+2 \log _{3} M$\ $=3 \log _{2} 4+2 \log _{3} 3=6+2=8$\