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doigiaythuytinh
ĐK:[TEX]0<x<1[/TEX]
[TEX]{\frac{log_{x^2} 4}{\sqrt {\frac{1}{6} + log_{x^6}(1-x)} - \sqrt{\frac{1}{2}} }\geq \frac{\sqrt6}{log_2(1-x) - log_4 x^4[/TEX]
\Leftrightarrow [TEX]\frac{log_x2}{{\sqrt{\frac{1}{6}+\frac{1}{6}.log_x{(1-x)}}- \sqrt{\frac{1}{2}}}} \geq \frac{\sqrt{6}}{log_2{(1-x)} - 2log_2x}[/TEX]
\Leftrightarrow[TEX]log_x{(1-x)} -2 \geq \sqrt{1+log_x{(1-x)}} - \sqrt{3}[/TEX]
Đặt: [TEX]t=\sqrt{1+log_x{(1-x)}}\Rightarrow log_x{(1-x)} = t^2-1[/TEX]BPT \Leftrightarrow [TEX]t^2-3 \geq t-\sqrt{3}[/TEX]
[TEX]{\frac{log_{x^2} 4}{\sqrt {\frac{1}{6} + log_{x^6}(1-x)} - \sqrt{\frac{1}{2}} }\geq \frac{\sqrt6}{log_2(1-x) - log_4 x^4[/TEX]
\Leftrightarrow [TEX]\frac{log_x2}{{\sqrt{\frac{1}{6}+\frac{1}{6}.log_x{(1-x)}}- \sqrt{\frac{1}{2}}}} \geq \frac{\sqrt{6}}{log_2{(1-x)} - 2log_2x}[/TEX]
\Leftrightarrow[TEX]log_x{(1-x)} -2 \geq \sqrt{1+log_x{(1-x)}} - \sqrt{3}[/TEX]
Đặt: [TEX]t=\sqrt{1+log_x{(1-x)}}\Rightarrow log_x{(1-x)} = t^2-1[/TEX]BPT \Leftrightarrow [TEX]t^2-3 \geq t-\sqrt{3}[/TEX]
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