\(A=\left(x+\frac{4}{9x}\right)+\left(y+\frac{4}{9y}\right)+\frac{5}{9}\left(\frac{1}{x}+\frac{1}{y}\right)\ge2\sqrt{x.\frac{4}{9x}}+2\sqrt{y.\frac{4}{9y}}+\frac{20}{9\left(x+y\right)}\)

\(\ge\frac{4}{3}+\frac{4}{3}+\frac{20}{12}=\frac{13}{3}\)

Dấu "=" xảy ra khi \(x=y=\frac{2}{3}\)

Tham khảo thanh này để soạn đề chính xác hơn nha :vvv

a) Ta có: \(M=\left(\dfrac{\sqrt{x}-3}{\sqrt{x}-2}-\dfrac{\sqrt{x}+1}{\sqrt{x}+3}\right)\cdot\dfrac{x+3\sqrt{x}}{7-\sqrt{x}}\)

\(=\left(\dfrac{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+3\right)}-\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+3\right)}\right)\cdot\dfrac{\sqrt{x}\left(\sqrt{x}+3\right)}{7-\sqrt{x}}\)

\(=\dfrac{x-9-\left(x-2\sqrt{x}+\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+3\right)}\cdot\dfrac{\sqrt{x}\left(\sqrt{x}+3\right)}{7-\sqrt{x}}\)

\(=\dfrac{x-9-x+\sqrt{x}+2}{\left(\sqrt{x}-2\right)}\cdot\dfrac{1}{-\left(\sqrt{x}-7\right)}\)

\(=\dfrac{\sqrt{x}-7}{\sqrt{x}-2}\cdot\dfrac{-1}{\sqrt{x}-7}\)

\(=\dfrac{-1}{\sqrt{x}-2}\)(1)

b) Ta có: \(x^2-4x=0\)

\(\Leftrightarrow x\left(x-4\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x-4=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=0\left(nhận\right)\\x=4\left(loại\right)\end{matrix}\right.\)

Thay x=0 vào biểu thức (1), ta được:

\(M=\dfrac{-1}{\sqrt{0}-2}=\dfrac{-1}{-2}=\dfrac{1}{2}\)

Vậy: Khi \(x^2-4x=0\) thì \(M=\dfrac{1}{2}\)

Ta có

\(A=4\left(a^2+b^2+c^2\right)\ge4\left(a+b+c\right)^2.\frac{1}{3}=3\)

1: A=2

=>\(\sqrt{x}+1=2\left(\sqrt{x}-2\right)\)

=>\(2\sqrt{x}-4=\sqrt{x}+1\)

=>\(\sqrt{x}=5\)

=>x=25

2: A<1

=>A-1<0

=>\(\dfrac{\sqrt{x}+1-\sqrt{x}+2}{\sqrt{x}-2}< 0\)

=>\(\dfrac{3}{\sqrt{x}-2}< 0\)

=>\(\sqrt{x}-2< 0\)

=>0<=x<4

3: A<1/3

=>A-1/3<0

=>\(\dfrac{\sqrt{x}+1}{\sqrt{x}-2}-\dfrac{1}{3}< 0\)

=>\(\dfrac{3\sqrt{x}+3-\sqrt{x}+2}{3\left(\sqrt{x}-2\right)}< 0\)

=>\(\dfrac{2\sqrt{x}+5}{3\left(\sqrt{x}-2\right)}< 0\)

=>\(\sqrt{x}-2< 0\)

=>0<=x<4

4:
A=căn x

=>\(\sqrt{x}+1=x-2\sqrt{x}\)

=>\(x-3\sqrt{x}-1=0\)

=>\(\left[{}\begin{matrix}\sqrt{x}=\dfrac{3+\sqrt{13}}{2}\left(nhận\right)\\\sqrt{x}=\dfrac{3-\sqrt{13}}{2}\left(loại\right)\end{matrix}\right.\)

=>\(x=\dfrac{11+3\sqrt{13}}{2}\)

a: Ta có: \(A=\left(\dfrac{2}{x-\sqrt{x}}-\dfrac{1}{\sqrt{x}-1}\right):\dfrac{x-4}{x\sqrt{x}+\sqrt{x}-2x}\)

\(=\dfrac{2-\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\dfrac{\sqrt{x}\left(x-2\sqrt{x}+1\right)}{x-4}\)

\(=\dfrac{-\sqrt{x}\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}+2\right)}\)

\(=\dfrac{-\sqrt{x}+1}{\sqrt{x}+2}\)

\(A=x^4-2x^2+1-3\left|x^2-1\right|-10\)

\(=\left|x^2-1\right|^2-3\left|x^2-1\right|-10\)

\(=\left(\left|x^2-1\right|-\frac{3}{2}\right)^2-\frac{49}{4}\ge-\frac{49}{4}\)

\(A_{min}=-\frac{49}{4}\) khi \(\left|x^2-1\right|=\frac{3}{2}\Rightarrow x=\pm\sqrt{\frac{5}{2}}\)

cảm ơn bạn

a) ĐKXĐ : \(3\le x\le7\)

Ta có \(A=1.\sqrt{x-3}+1.\sqrt{7-x}\)

\(\le\sqrt{\left(1+1\right)\left(x-3+7-x\right)}=\sqrt{8}\)(BĐT Bunyacovski)

Dấu "=" xảy ra <=> \(\dfrac{1}{\sqrt{x-3}}=\dfrac{1}{\sqrt{7-x}}\Leftrightarrow x=5\)

Max và min chứ có ngu đến mức k bt lm cái đó đâu