Ta thấy: \(a+b\le1\Leftrightarrow\hept{\begin{cases}a\le1-b\\b\le1-a\end{cases}}\Leftrightarrow\hept{\begin{cases}1+a\le2-b\\1+b\le2-a\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}\frac{a}{1+b}\ge\frac{a}{2-a}\\\frac{b}{1+a}\ge\frac{b}{2-b}\end{cases}}\Rightarrow\frac{a}{1+b}+\frac{b}{1+a}\ge\frac{a}{2-a}+\frac{b}{2-b}\)

\(\Rightarrow S=\frac{a}{1+b}+\frac{b}{1+a}+\frac{1}{a+b}\ge\frac{a}{2-a}+\frac{b}{2-b}+\frac{1}{a+b}\)

\(=\frac{2}{2-a}-1+\frac{2}{2-b}-1+\frac{1}{a+b}=\frac{2}{2-a}+\frac{2}{2-b}+\frac{1}{a+b}-2\)

\(=2\left(\frac{1}{2-a}+\frac{1}{2-b}+\frac{1}{2\left(a+b\right)}-1\right)\)

Áp dụng bất đẳng thức sau: \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\ge\frac{9}{x+y+z}\)

\(\Rightarrow\frac{1}{2-a}+\frac{1}{2-b}+\frac{1}{2\left(a+b\right)}\ge\frac{9}{4-\left(a+b\right)+2\left(a+b\right)}=\frac{9}{4+a+b}\)

Lại có: \(a+b\le1\Rightarrow4+a+b\le5\Rightarrow\frac{9}{4+a+b}\ge\frac{9}{5}\)

\(\Rightarrow\frac{1}{2-a}+\frac{1}{2-b}+\frac{1}{2\left(a+b\right)}\ge\frac{9}{5}\Leftrightarrow2\left(\frac{1}{2-a}+\frac{1}{2-b}+\frac{1}{2\left(a+b\right)}-1\right)\ge\frac{8}{5}\)

\(\Rightarrow S\ge\frac{8}{5}.\)

Vậy \(Min_S=\frac{8}{5}.\)Dấu "=" xảy ra khi \(a=b=\frac{2}{5}.\)

1x+1y≥4x+y,x,y>01x+1y≥4x+y,x,y>0.

Dấu "=" xảy ra $\iff x=y$. ( Chứng minh bằng phương pháp biến tổi tươg đuơng)

M=1a2+b2+2ab+4abM=1a2+b2+2ab+4ab $=\left(\frac{1}{a^2+b^2}+\frac{1}{2ab}\right)+\left(4ab+\frac{1}{4ab}\right)+\frac{5}{4ab}$ +) $4ab+\frac{1}{4ab}\ge 2\sqrt{4ab\cdot \frac{1}{4ab}}=2$ +) 

\(M=\frac{1}{a^2+b^2}+\frac{2}{ab}+4ab\)

\(=\frac{1}{a^2+b^2}+\frac{1}{2ab}+\frac{1}{4ab}+4ab+\frac{5}{4ab}\)

\(\ge\frac{4}{\left(a+b\right)^2}+2\sqrt{\frac{1}{4ab}.4ab}+\frac{5}{4ab}\)

( Nếu đi thi thì sẽ phải chứng minh \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\) cái này nhân chéo và cô si là xong )

Ta có BĐT phụ: \(\left(a+b\right)^2\ge4ab\)

\(\Leftrightarrow\left(a-b\right)^2\ge0\)( đúng )

\(\Rightarrow M\ge\frac{4}{1}+2+5=11\)

Dấu "=" xảy ra <=> a=b=1/2 

Vậy ...

\(2a^2+\frac{1}{a^2}+\frac{b^2}{4}=4\Leftrightarrow\left(a^2+\frac{1}{a^2}-2\right)+\left(a^2+\frac{b^2}{4}-ab\right)=4-ab-2\)

\(\Leftrightarrow\left(a-\frac{1}{a}\right)^2+\left(a-\frac{b}{2}\right)^2=2-ab\)

\(VF=2-ab=\left(a-\frac{1}{a}\right)^2+\left(b-\frac{b}{2}\right)^2\ge0\)

Hay \(ab\le2\)

Dấu "=" xảy ra khi \(\hept{\begin{cases}a=\frac{1}{a}\\b=\frac{b}{2}\end{cases}}\Leftrightarrow\orbr{\begin{cases}\left(a;b\right)=\left(1;\frac{1}{2}\right)\\\left(a;b\right)=\left(-1;-\frac{1}{2}\right)\end{cases}}\)

ủa bạn tìm giá trị nhỏ nhất của biểu thức S=ab+2019 mà 

\(S=\frac{a^2}{a+ab}+\frac{b^2}{b+ab}+\frac{1}{a+b}\ge\frac{\left(a+b\right)^2}{a+b+2ab}+\frac{1}{a+b}\ge\frac{\left(a+b\right)^2}{a+b+\frac{\left(a+b\right)^2}{2}}+\frac{1}{a+b}\ge\frac{1}{1+\frac{1}{2}}+1=\frac{5}{3}\)

\(\Rightarrow S_{min}=\frac{5}{3}\) khi \(a=b=\frac{1}{2}\)

Tìm trên mạng ý

\(a+\frac{1}{b}\le1=>ab+1\le b\)

\(b\le ab+1\ge2\sqrt{ab}=>\sqrt{b}\ge2\sqrt{a}=>\frac{b}{a}\ge4\)

\(T=\frac{ab}{a^2+b^2}=\frac{1}{\frac{a}{b}+\frac{b}{a}}=\frac{1}{\frac{a}{b}+\frac{b}{16a}+\frac{15b}{16a}}\)

áp dụng cô si 

\(\frac{a}{b}+\frac{b}{16a}\ge2\sqrt{\frac{ab}{16ab}}=\frac{1}{2}=>T\le\frac{1}{\frac{1}{2}+\frac{15}{16}.4}=\frac{4}{17}\)

\(=>MaxT=\frac{4}{17}\)

dấu = xảy ra khi

\(b=4a;\frac{a}{b}=\frac{b}{16a};ab=1\)

\(=>\hept{\begin{cases}4a^2=1\\b=4a\end{cases}=>\hept{\begin{cases}a=\frac{1}{2}\\b=2\end{cases}}}\)