JEE Advanced 2020 Paper 1Question 1
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 $(\log_3x_1+\log_3x_2+\log_3x_3)$, where $x_1,x_2,x_3$ are positive real numbers for which $x_1+x_2+x_3=9$. Then the value of $\log_2(m^3)+\log_3(M^2)$ is ______
Sets, Relations and Function
Answer$8$
Solution
By AM-GM $3^{y_1}+3^{y_2}+3^{y_3}\ge3\cdot3^{3}=81$, so $m=\log_381=4$. Also $x_1x_2x_3\le27$, so $M=3$. Then $\log_2 4^3+\log_3 3^2=6+2=8$.