\(\dfrac{a}{b}+\dfrac{b}{a}\ge2\Leftrightarrow\dfrac{a}{b}+\dfrac{b}{a}-2\ge0\)

\(\Leftrightarrow\dfrac{a^2+b^2-2ab}{ab}\ge0\)

\(\Leftrightarrow\dfrac{\left(a-b\right)^2}{ab}\ge0\left(ab>0\right)\)

Ta có: a,b > 0

=> \(\dfrac{a}{b},\dfrac{b}{a}>0\)

=> \(\dfrac{a}{b}+\dfrac{b}{a}\ge2\sqrt{\dfrac{a}{b}.\dfrac{b}{a}}=2\)

Bài làm :

Ta có : \(\left(x-y\right)^2\ge0\)

\(\Rightarrow x^2+y^2\ge2xy\)

\(\Rightarrow\left(x+y\right)^2\ge4xy\)

\(\Rightarrow\dfrac{1}{xy}\ge\dfrac{4}{\left(x+y\right)^2}\left(1\right)\)

Áp dụng BĐT (1) ta có :

\(\dfrac{a}{b+c}+\dfrac{c}{d+a}=\dfrac{a^2+ad+bc+c^2}{\left(b+c\right)\left(d+a\right)}\ge\dfrac{4\left(a^2+ad+bc+c^2\right)}{\left(a+b+c+d\right)^2}\left(2\right)\)

Tương tự : \(\dfrac{b}{c+d}+\dfrac{d}{a+b}\ge\dfrac{4\left(b^2+ab+cd+d^2\right)}{\left(a+b+c+d\right)^2}\left(3\right)\)

Cộng các về của các BĐT (2) và (3) ta được :

\(\dfrac{a}{b+c}+\dfrac{b}{c+d}+\dfrac{c}{d+a}+\dfrac{d}{a+b}\ge\dfrac{4\left(a^2+b^2+c^2+d^2+ad+bc+ab+cd\right)}{\left(a+b+c+d\right)^2}\)

\(\dfrac{a}{b+c}+\dfrac{b}{c+d}+\dfrac{c}{d+a}+\dfrac{d}{a+b}\ge\dfrac{2\left(2a^2+2b^2+2c^2+2d^2+2ad+2bc+2ab+2cd\right)}{\left(a+b+c+d\right)^2}\)

\(\dfrac{a}{b+c}+\dfrac{b}{c+d}+\dfrac{c}{d+a}+\dfrac{d}{a+b}\ge\dfrac{2[\left(a+b\right)^2+\left(b+c\right)^2+\left(c+d\right)^2+\left(a+d\right)^2]}{\left(a+b+c+d\right)^2}=2B\)

Ta dễ dàng chứng minh được : \(B\ge1\)

Thật vậy :

\(\dfrac{\left(a+b\right)^2+\left(b+c\right)^2+\left(c+d\right)^2+\left(a+d\right)^2}{\left(a+b+c+d\right)^2}\ge1\)

\(\Leftrightarrow\left(a+b\right)^2+\left(b+c\right)^2+\left(c+d\right)^2+\left(d+a\right)^2\ge\left(a+b+c+d\right)^2\)

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

\(\Rightarrowđpcm\)

Dấu đằng thức xảy ra : \(\Leftrightarrow a=c;b=d\)

khó thế tui ko hỉu

Ta có BĐT : a² + b² ≥ 2ab

=> \(\dfrac{a^2+b^2}{ab}\) ≥ 2

=> \(\dfrac{a}{b}+\dfrac{b}{a}\) ≥ a

Dấu " = " xảy ra khi : a = b

\(\text{ Ta có : }\dfrac{b}{a}+\dfrac{a}{b}=\dfrac{b^2}{ab}+\dfrac{a^2}{ab}\\ \\ =\dfrac{a^2+b^2}{ab}\)

Áp dụng BDT Cô-si: \(x^2+y^2\ge2xy\)

\(\Rightarrow\dfrac{b}{a}+\dfrac{a}{b}=\dfrac{a^2+b^2}{ab}\ge\dfrac{2ab}{ab}\ge2\left(đpcm\right)\)

Vậy \(\dfrac{b}{a}+\dfrac{a}{b}\ge2\). Đẳng thức xảy ra khi \(a=b\)

\(\Leftrightarrow\dfrac{a}{\sqrt{4b^2+bc+4c^2}}+\dfrac{b}{\sqrt{4c^2+ca+4a^2}}+\dfrac{c}{\sqrt{4a^2+ab+4b^2}}\ge1\)

Ta có:

\(\sum\left(\dfrac{a}{\sqrt{4b^2+bc+4c^2}}\right)^2\sum a\left(4b^2+bc+4c^2\right)\ge\left(a+b+c\right)^3\)

Nên ta chỉ cần chứng minh:

\(\dfrac{\left(a+b+c\right)^3}{a\left(4b^2+bc+4c^2\right)+b\left(4c^2+ac+4a^2\right)+c\left(4a^2+ab+4b^2\right)}\ge1\)

\(\Leftrightarrow\dfrac{\left(a+b+c\right)^3}{4a\left(b^2+c^2\right)+4b\left(c^2+a^2\right)+4c\left(a^2+b^2\right)+3abc}\ge1\)

\(\Leftrightarrow a^3+b^3+c^3+3abc\ge ab\left(a+b\right)+bc\left(b+c\right)+ca\left(c+a\right)\) (đúng theo Schur bậc 3)

a) theo định lý côsi :

\(\dfrac{a}{b}\)+\(\dfrac{b}{a}\)luôn >=2 với mọi a, b , a.b > 0

Chứng minh: \(x^3+y^3\ge xy\left(x+y\right)\left(1\right)\)

\(x^3+y^3\ge xy\left(x+y\right)\)

\(\Leftrightarrow\left(x+y\right)^3-3xy\left(x+y\right)\ge xy\left(x+y\right)\)

\(\Leftrightarrow\left(x+y\right)^3\ge4xy\left(x+y\right)\)

\(\Leftrightarrow\left(x+y\right)^2\ge4xy\)

\(\Leftrightarrow\left(x-y\right)^2\ge0\) đúng

\(\Rightarrow\left(1\right)\) đúng

Áp dụng BĐT \(x^3+y^3\ge xy\left(x+y\right)\)

\(\dfrac{a^3+b^3}{ab}+\dfrac{b^3+c^3}{bc}+\dfrac{c^3+a^3}{ca}\)

\(\ge\dfrac{ab\left(a+b\right)}{ab}+\dfrac{bc\left(b+c\right)}{bc}+\dfrac{ca\left(c+a\right)}{ca}\)

\(=2\left(a+b+c\right)\)

Đặt \(P=\dfrac{a}{b+c}+\dfrac{b}{c+d}+\dfrac{c}{a+d}+\dfrac{d}{a+b}\)

\(P=\dfrac{a^2}{ab+ac}+\dfrac{b^2}{bc+bd}+\dfrac{c^2}{ac+cd}+\dfrac{d^2}{ad+bd}\)

\(P\ge\dfrac{\left(a+b+c+d\right)^2}{ab+2ac+bc+2bd+cd+ad}=\dfrac{\left(a+c\right)^2+\left(b+d\right)^2+2\left(a+c\right)\left(b+d\right)}{2ac+2bd+ab+bc+cd+ad}\)

\(P\ge\dfrac{4ac+4bd+2ab+2bc+2cd+2ad}{2ac+2bd+ab+bc+cd+ad}=2\) (đpcm)

Dấu "=" xảy ra khi \(a=b=c=d\)

Lời giải:

Với $a,b,c>0$ ta có:

$M> \frac{a}{a+b+c}+\frac{b}{b+c+a}+\frac{c}{c+a+b}=\frac{a+b+c}{a+b+c}{a+b+c}=1(*)$

Mặt khác:
Xét hiệu: $\frac{a}{a+b}-\frac{a+c}{a+b+c}=\frac{-bc}{(a+b)(a+b+c)}<0$ với mọi $a,b,c>0$

$\Rightarrow \frac{a}{a+b}< \frac{a+c}{a+b+c}$

Tương tự ta cũng có: $\frac{b}{b+c}< \frac{b+a}{a+b+c}; \frac{c}{c+a}< \frac{c+b}{a+b+c}$

Cộng lại ta được: $M< \frac{a+c+b+a+c+b}{a+b+c}=\frac{2(a+b+c)}{a+b+c}=2(**)$

Từ $(*); (**)\Rightarrow 1< M< 2$ nên $M$ không là số nguyên.