Áp dụng BĐT Cauchy-Schwarz dạng phân thức cho các số không âm:

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{9}{a+b+c}\)

\(''=''\Leftrightarrow a=b=c\)

Trình bày như vậy khó lắm nếu bn ấy chưa tìm hiểu

BĐT

\(\Leftrightarrow\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=9\)( do a,b,c>0)

\(\Leftrightarrow\left(\frac{a}{b}-2+\frac{b}{a}\right)+\left(\frac{b}{c}-2+\frac{c}{b}\right)+\left(\frac{a}{c}-2+\frac{c}{a}\right)\ge0\)

\(\Leftrightarrow\frac{\left(a-b\right)^2}{ab}+\frac{\left(b-c\right)^2}{bc}+\frac{\left(a-c\right)^2}{ac}\ge0\)(đúng)

BĐT \(\Leftrightarrow\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9\)

Áp dụng bđt Cô-si :

\(a+b+c\ge3\sqrt[3]{abc}\)

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge3\sqrt[3]{\frac{1}{abc}}\)

Nhân theo vế của 2 bđt :

\(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge3\sqrt[3]{abc}\cdot\frac{3}{\sqrt[3]{abc}}=9\)

Dấu "=" xảy ra \(\Leftrightarrow a=b=c\)

\(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9.\)

\(a+b+c\ge3\sqrt[3]{abc}\)     1

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge3\sqrt[3]{\frac{1}{abc}}\)  2

nhân 1 vs 2 

\(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9\sqrt[3]{\frac{abc}{abc}}=9\)

\(\frac{a}{b^2}+\frac{1}{a}\ge\frac{2}{b}\) BĐT Cô-si

Tương tự suy ra đpcm

Áp dụng bất đẳng thức Cô-si ta có:

\(\dfrac{a^2}{b^3}+\dfrac{1}{a}+\dfrac{1}{a}\ge\sqrt[3]{\dfrac{a^2}{b^3}.\dfrac{1}{a}.\dfrac{1}{a}}=\dfrac{3}{b}\)

\(\dfrac{c^2}{a^3}+\dfrac{1}{c}+\dfrac{1}{c}\ge\sqrt[3]{\dfrac{c^2}{a^3}.\dfrac{1}{c}.\dfrac{1}{c}}=\dfrac{3}{a}\)

\(\dfrac{c^2}{a^3}+\dfrac{1}{c}+\dfrac{1}{c}\ge\sqrt[3]{\dfrac{c^2}{a^3}.\dfrac{1}{c}.\dfrac{1}{c}}=\dfrac{3}{a}\)

Cộng theo vế ta được:

\(\dfrac{a^2}{b^3}+\dfrac{b^2}{c^3}+\dfrac{a^2}{a^3}+\dfrac{2}{a}+\dfrac{2}{b}+\dfrac{2}{c}\ge3\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\)

\(\Leftrightarrow\dfrac{a^2}{b^3}+\dfrac{b^2}{c^3}+\dfrac{c^2}{a^3}\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{3}{\sqrt[3]{abc}}\)

Ta cần chứng minh \(\frac{3}{\sqrt[3]{abc}}\ge\frac{9}{abc+2}\Leftrightarrow abc+2\ge3\sqrt[3]{abc}\)

BĐT trên luôn đúng theo AM-GM vì: \(abc+2=abc+1+1\ge3\sqrt[3]{abc}\)

Dấu "=" xảy ra khi \(a=b=c=1\)

bài hơi khoai

Không mất tính tổng quát giả sử \(c=max\left\{a,b,c\right\}\)

\(\Rightarrow2c\ge a+b\)

\(\Rightarrow c\ge\frac{a+b}{2}\)

Từ giả thiết \(\Rightarrow a,b\le1\)

\(\Rightarrow ab\le1\)( *)

Đặt \(P=\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}-\frac{5}{2}\)

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

Đặt \(S=\frac{1}{a+b+\frac{1}{a+b}}+a+b+\frac{1}{a+b}-\frac{5}{2}\)

Xét hiệu \(P-S=\)\(\frac{1}{a+b}+\frac{1}{b+\frac{1-ab}{a+b}}+\frac{1}{a+\frac{1-ab}{a+b}}-\frac{5}{2}-\)\(-\frac{1}{a+b+\frac{1}{a+b}}-a-b-\frac{1}{a+b}+\frac{5}{2}\)

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

\(=\frac{a+b}{b^2+1}+\frac{a+b}{c^2+1}-\left(a+b\right)\left[1+\frac{1}{1+\left(a+b\right)^2}\right]\)

Ta sẽ chứng minh \(\frac{a+b}{b^2+1}+\frac{a+b}{c^2+1}-\left(a+b\right)\left[1+\frac{1}{1+\left(a+b\right)^2}\right]\ge0\)

\(\Leftrightarrow\frac{a+b}{b^2+1}+\frac{a+b}{c^2+1}\ge\left(a+b\right)\left[1+\frac{1}{1+\left(a+b\right)^2}\right]\)

\(\Leftrightarrow\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge1+\frac{1}{1+\left(a+b\right)^2}\)

\(\Leftrightarrow\frac{2+a^2+b^2}{\left(1+a^2\right)\left(1+b^2\right)}\ge\frac{2+\left(a+b\right)^2}{1+\left(a+b\right)^2}\)

\(\Rightarrow\left(2+b^2+a^2\right)\left[1+\left(a+b\right)^2\right]\ge\left[2+\left(a+b\right)^2\right]\left(1+a^2\right)\left(1+b^2\right)\)

\(\Leftrightarrow2+2\left(a+b\right)^2+\left(a+b\right)^2\left(a^2+b^2\right)+a^2+b^2\ge\left[2+\left(a+b\right)^2\right]\left(1+a^2+b^2+a^2b^2\right)\)

\(\Leftrightarrow2+2\left(a+b\right)^2+\left(a+b\right)^2\left(a^2+b^2\right)+a^2+b^2-2a^2b^2-\left(a+b\right)^2\left(a^2+b^2\right)-\left(a+b\right)^2a^2b^2\)\(-2-2\left(a^2+b^2\right)-\left(a+b^2\right)\ge0\)

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

\(\Leftrightarrow ab\left[ab\left(a+b\right)^2+2ab-2\right]\le0\)

\(\Leftrightarrow ab\left(a+b\right)^2+2ab-2\le0\)( do a,b \(\ge0\))

\(\Leftrightarrow ab\left(a+b\right)^2\le2\left(1-ab\right)\)

\(\Leftrightarrow ab\left(a+b\right)^2\le2c\left(a+b\right)\) (1)

Mà \(c\ge\frac{a+b}{2}\)

\(\Rightarrow2c\left(a+b\right)\ge\left(a+b\right)^2\)

Ta có: \(\left(a+b\right)^2\ge ab\left(a+b\right)^2\)

\(\Leftrightarrow\left(a+b\right)^2\left(1-ab\right)\ge0\)( đúng do (*) ) 

\(\Rightarrow\left(1\right)\)đúng

\(\Rightarrow P-S\ge0\)

\(\Rightarrow P\ge S\)

Ta phải chứng minh \(S\ge0\)

\(\Leftrightarrow\frac{1}{a+b+\frac{1}{a+b}}+a+b+\frac{1}{a+b}\ge\frac{5}{2}\)

\(\Leftrightarrow\frac{a+b}{1+\left(a+b\right)^2}+\frac{1+\left(a+b\right)^2}{a+b}\ge\frac{5}{2}\) (2) 

Đặt \(x=\frac{1+\left(a+b\right)^2}{a+b}\)

Ta có: \(1+\left(a+b\right)^2\ge2\left(a+b\right)\)

\(\Leftrightarrow\left(a+b-1\right)^2\ge0\)( đúng )

\(\Rightarrow x=\frac{1+\left(a+b\right)^2}{a+b}\ge2\)

=> (2) có dạng \(x+\frac{1}{x}\ge\frac{5}{2}\)

\(\Leftrightarrow2x^2-5x+2\ge0\)

\(\Leftrightarrow\left(2x-1\right)\left(x-2\right)\ge0\)( đúng )

\(\Rightarrow S\ge0\)mà \(P\ge S\)

\(\Rightarrow P\ge0\)

\(\Leftrightarrow\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\frac{5}{2}\)

Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}a+b=1\\ab+bc+ca=1\\ab\left[ab\left(a+b\right)^2+2ab-2\right]=0\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}a=c=1;b=0\\b=c=1;a=0\end{cases}}\)

sửa một chút là cái dòng thứ 4 là từ giả thiết\(\Rightarrow0\le a,b\le1\)

a) \(\left(a+b\right)^2\ge4ab\)

\(\Leftrightarrow\frac{a+b}{ab}\ge\frac{4}{a+b}\)

\(\Leftrightarrow\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\left(đpcm\right)\)

Áp dụng BĐT Cô -si cho 3 số dương:

\(a+b+c\ge3\sqrt[3]{abc};\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge3\sqrt[3]{\frac{1}{abc}}\)

\(\Rightarrow\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9\)

\(\Leftrightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{9}{a+b+c}\)

vì sao \(\left(a+b\right)^2\ge4ab\)