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zmudz
 one year ago
For x, y, and z positive real numbers, what is the maximum possible value for
\(\sqrt{\frac{3x+4y}{6x+5y+4z}}
+
\sqrt{\frac{y+2z}{6x+5y+4z}}
+
\sqrt{\frac{2z+3x}{6x+5y+4z}}?
\)
Also, find what z/x is if (x,y,z) achieves the maximum value.
Thank you! I am so stuck right now.
zmudz
 one year ago
For x, y, and z positive real numbers, what is the maximum possible value for \(\sqrt{\frac{3x+4y}{6x+5y+4z}} + \sqrt{\frac{y+2z}{6x+5y+4z}} + \sqrt{\frac{2z+3x}{6x+5y+4z}}? \) Also, find what z/x is if (x,y,z) achieves the maximum value. Thank you! I am so stuck right now.

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ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Let \(a=\sqrt{\frac{3x+4y}{6x+5y+4z}}\\ b=\sqrt{\frac{y+2z}{6x+5y+4z}}\\ c=\sqrt{\frac{2z+3x}{6x+5y+4z}}\) Firstly, notice that \(a^2+b^2+c^2=1\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3appealing to cauchy schwarz inequality gives \[a*1+b*1+c*1\le \sqrt{(a^2+b^2+c^2)(1^2+1^2+1^2)}=\sqrt{3}\]
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