Weighted Harmonic Mean

Q. The minimum value of $2^{\sin x}+2^{\cos x}$ is

1. $2^{1-\sqrt{2}}$
2. $2^{1-\frac{1}{\sqrt{2}}}$
3. $2^{-1+\sqrt{2}}$
4. $2^{-1+\frac{1}{\sqrt{2}}}$

Q.

Let $m$be the minimum possible value of ${\log }_3(3^{y_1}+3^{y_2}+3^{y_3})$ 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 $({\log }_{3}1+{\log }_{3}2+{\log }_{3}3),$ where$x_{1,2,3}$ are positive real numbers for which $x_1+x_2+x_3=9.$Then the value of $lo{g}_2(3)+{\log }_3(2)$ is _____

Q. Let $m$ be the minimum possible value of ${\log }_3(3^{y_1}+3^{y_2}+3^{y_3})$, 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 $({\log }_3{x}_1+{\log }_3{x}_2+{\log }_3{x}_3),$ 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