JEE Advanced 2020 · Paper 1 · Mathematics Q13

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MathematicsNumerical+4Question 49 of 54

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 $(\log_3 x_1 + \log_3 x_2 + \log_3 x_3)$ , where $x_1, x_2, x_3$ are positive real numbers for which $x_1 + x_2 + x_3 = 9$ . Then the value of $\log_2(m^3) + \log_3(M^2)$ is ______

Your answer

Answer and worked solution

Official answer: 8

Minimum $m$ . By AM-GM,
$3^{y_1}+3^{y_2}+3^{y_3}\ge3\cdot3^{(y_1+y_2+y_3)/3}=3\cdot27=81,$
with equality at $y_i=3$ . So $m=\log_381=4$ .

Maximum $M$ . By AM-GM, $x_1x_2x_3\le\left(\frac93\right)^3=27$ , so $M=\log_327=3$ .

Value.
$\log_2(4^3)+\log_3(3^2)=6+2=8.$
Answer: 8

Topics:AM-GM-HM inequalities Logarithms

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JEE Advanced 2020 · Paper 1, Mathematics question 13