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

Question

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 $\log_{2} (m^{3}) + \log_{3} (M^{2})$ is _____ . [2020]

Smart Achievers · Accepted Answer

(8)

By AM--GM inequality

$AM \geq GM$

$∴$ $\frac{3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}}}{3} \geq {[3^{(y_{1} + y_{2} + y_{3})}]}^{\frac{1}{3}}$

$\Rightarrow 3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}} \geq 3^{4}$

$\Rightarrow \log_{3} (3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}}) \geq 4 \Rightarrow m = 4$

$∵$ $\log_{3} x_{1} + \log_{3} x_{2} + \log_{3} x_{3} = \log_{3} (x_{1} x_{2} x_{3})$

$Again by AM-GM inequality$

$AM \geq GM$

$\frac{x_{1} + x_{2} + x_{3}}{3} \geq \sqrt[3]{x_{1} x_{2} x_{3}} \Rightarrow x_{1} x_{2} x_{3} \leq 27$

$\Rightarrow \log_{3} (x_{1} x_{2} x_{3}) \leq \log_{3} (3^{3})$

$\Rightarrow \log_{3} x_{1} + \log_{3} x_{2} + \log_{3} x_{3} \leq 3 \Rightarrow M = 3$

$Now, \log_{2} (m^{3}) + \log_{3} (M^{2}) = 6 + 2 = 8$

Solution

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Solution

Q. Let m be the minimum possible value of log3(3y1+3y2+3y3), where y1,y2,y3 are real numbers for which y1+y2+y3=9. Let M be the maximum possible value of (log3x1+log3x2+log3x3), where x1,x2,x3 are positive real numbers for which x1+x2+x3=9. Then the value of log2(m3)+log3(M2) is _____ . [2020]

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