Xét các số thực dương x, y

A

- Hướng dẫn giải

Đáp án A

$$\begin{gathered} \ln \left(\frac{1-2x}{x+y}\right)=3x+y-1, \left(0<x<\frac{1}{2},y>0\right)\iff \ln \left(1-2x\right)+1-2x=\ln (x+y)+x+y \\ f\left(t\right)=t+\ln t\Rightarrow f\text{'}\left(t\right)=1+\frac{1}{t}>0 \\ \Rightarrow f\left(1-2x\right)=f(x+y)\iff 1-2x=x+y\iff y=1-3x \\ P=\frac{1}{x}+\frac{1}{\sqrt{xy}}=\frac{1}{x}+\frac{1}{\sqrt{x\left(1-3x\right)}} \\ \Rightarrow P\text{'}=-\frac{1}{x^2}+\frac{6x-1}{2\sqrt{x\left(1-3x\right)} x(1-3x)}=\frac{-2\sqrt{x\left(1-3x\right)} (1-3x)+(6x-1)x}{2\sqrt{x\left(1-3x\right)} x^2(1-3x)} \\ P\text{'}=0\iff 2\sqrt{x\left(1-3x\right)}(1-3x)=6x^2-x \\ \iff \left\{\begin{array}{l}6x^2-x>0\\ 4x{(1-3x)}^3={\left(6x2-x\right)}^2\end{array}\iff \left\{\begin{array}{l}[\begin{array}{c}x<0\\ x>\frac{1}{6}\end{array}\\ 4x-36x^2+108x^3-108x^4=26x^4-12x^3+x^2\end{array}\right.\right. \\ \iff x=y=\frac{1}{4} \\ \Rightarrow P=8 \end{gathered}$$