Đề là \(M=\frac{x^2+y^2+3}{x+y+1}\) à bạn ??

Không bạn, đề như mình ý

Thế này hả:

\(M=x^2+y^2+\frac{3}{x+y+1}\)

Phải vậy ko Ngọc Hạnh Phan 

\(y=\frac{x}{1-x}+\frac{5}{x}=\frac{x}{1-x}+\frac{5\left(1-x\right)}{x}+5\)

Áp dụng BĐT Cô - si ta có :

\(\frac{x}{1-x}+\frac{5\left(1-x\right)}{x}\ge2\sqrt{\frac{5x\left(1-x\right)}{x\left(1-x\right)}}=2\sqrt{5}\)

\(\Rightarrow y\ge5+2\sqrt{5}\)

Dấu \("="\) xảy ra khi \(x^2=5x^2-10x+5\Leftrightarrow4x^2-10x+5=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x_1=\frac{5+\sqrt{5}}{4}\\x_2=\frac{5-\sqrt{5}}{4}\end{matrix}\right.\)

x - 921 = 1960

x = 1960 + 921

x = 2881

X - 8289 : 9 = 1960

X - 921= 1960

X = 1960 + 921

X = 2881

\(y=\frac{x}{3}+\frac{5}{2x-1}=\frac{2x}{6}+\frac{5}{2x-1}=\frac{2x-1}{6}+\frac{5}{2x-1}+\frac{1}{6}\)

\(\Rightarrow y\ge2\sqrt{\frac{2x-1}{6}.\frac{5}{2x-1}}+\frac{1}{6}=\frac{\sqrt{30}}{3}+\frac{1}{6}\)

\(\Rightarrow P_{min}=\frac{\sqrt{30}}{3}+\frac{1}{6}\)

Dấu "=" xảy ra khi \(\left(2x-1\right)^2=30\Rightarrow x=\frac{\sqrt{30}+1}{2}\)