Ta có: \(\frac{2a^2+3b^2}{2a^3+3b^3}\left(a+b\right)=1+ab\frac{2a+3b}{2a^3+3b^3}\)

Áp dụng BĐT Holder ta có: 

\(\left(2a^3+3b^3\right)\left(2+3\right)^2\ge\left(2a+3b\right)^3\)

Vậy ta có thể viết lại BĐT cần chứng minh như sau;

\(VT\left(a+b\right)\le2+25ab\left(\frac{1}{\left(2a+3b\right)^2}+\frac{1}{\left(2b+3a\right)^2}\right)\)

Nó đủ để ta có thể thấy rằng 

\(25ab\left[\left(2b+3a\right)^2+\left(2a+3b\right)^2\right]\le2\left(2a+3b\right)^2\left(2b+3a\right)^2\)

\(\Leftrightarrow59\left(a^2-b^2\right)^2+13\left(a^4+b^4-a^3b-ab^3\right)\ge0\)

BĐT cuối cùng đúng nên ta có ĐPCM

ok jjj

Đặt \(\frac{a}{b}=t\)do a>0, b>0 nên t>0

Khi đó BĐT \(\frac{2a^2+3b^2}{2a^3+3b^3}+\frac{2b^2}{3b^3}+\frac{2b^2+3a^2}{2b^3+3a^2}\le\frac{4}{a+b}\left(1\right)\)trở thành

\(\frac{2t^2+3}{2t^3+3}+\frac{2+3t^2}{3+3t^3}\le\frac{4}{t+1}\)

\(\Leftrightarrow\left(2t^2+3\right)\left(2+3t^2\right)\left(t+1\right)+\left(2+3t^2\right)\left(2t^2+1\right)\left(t+1\right)\le4\left(2t^3+3\right)\left(2+3t^2\right)\)

\(\Leftrightarrow\left(t+1\right)\left(12t^5+13t^3+13t^2+12\right)\le4\left(6t^6+13t^3+6\right)\)

\(\Leftrightarrow12\left(t^6-t^5-t+1\right)-13t^2\left(t^2-12t+1\right)\ge0\)

\(\Leftrightarrow12\left(t-1\right)^2\left[12\left(t^4+t^3+t^2+t+1\right)-13t^2\right]\ge0\)

\(\Leftrightarrow\left(t-1\right)^2\left[12\left(t^4+t^3+t^2+t+1\right)-13t^2\right]\ge0\left(2\right)\)

Ta có \(12\left(t^4+t^3+t^2+t+1\right)-13t^2=12t^4+12t\left(t-1\right)^2+23t^2+12>0\forall t>0\)

BĐT (2) đúng với mọi t>0

=> BĐT (1) đúng với mọi a,b>0

Dấu "=" xảy ra <=> t=1 <=> a=b

\(\Leftrightarrow\frac{\left(2a^2+3b^2\right)\left(a+b\right)}{2a^3+3b^3}+\frac{\left(2b^2+3a^2\right)\left(a+b\right)}{2b^3+3a^3}\le4\)

\(\Leftrightarrow\frac{2a^3+3b^3+2a^2b+3ab^2}{2a^3+3b^3}+\frac{2b^3+3a^3+2ab^2+3ab^2}{2b^3+3a^3}\le4\)

\(\Leftrightarrow\frac{2a^2b+3ab^2}{2a^3+3b^3}+\frac{2ab^2+3ab^2}{2b^3+3a^3}\le2\)

\(\Leftrightarrow\frac{2\left(\frac{a}{b}\right)^2+3\left(\frac{a}{b}\right)}{2\left(\frac{a}{b}\right)^3+3}+\frac{2\left(\frac{a}{b}\right)+3\left(\frac{a}{b}\right)^2}{3\left(\frac{a}{b}\right)^3+2}\le2\)

Đặt \(\frac{a}{b}=x>0\Rightarrow\frac{2x^2+3x}{2x^3+3}+\frac{3x^2+2x}{3x^3+2}\le2\)

\(\Leftrightarrow\left(x-1\right)^2\left(12x^4+12x^3-x^2+12x+12\right)\ge0\) (luôn đúng)

Dấu "=" xảy ra khi \(x=1\) hay \(a=b\)

Hơi trâu bò :D

\(\dfrac{4}{a+b}-\dfrac{2a^2+3b^2}{2a^3+3b^3}-\dfrac{2b^2+3a^2}{2b^3+3a^3}=\dfrac{\left(a-b\right)^2.\left(12b^4+12ab^3-a^2b^2+12a^3b+12a^4\right)}{\left(a+b\right)\left(2a^3+3b^3\right)\left(2b^3+3a^3\right)}\ge0\)

PS: Còn cách dùng holder nữa mà lười quá

holder Câu hỏi của Lê Minh Đức - Toán lớp 9 - Học toán với OnlineMath

Bài này dễ mà bạn

dễ thì bn giải hộ mk đi,nói đc lm đc nhỉ

Ta CM BĐT phụ sau: \(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)

Ta có: \(\frac{1}{a}+\frac{1}{b}\ge\frac{2}{\sqrt{ab}},a+b\ge2\sqrt{ab}\)( co si với a,b>0)

Suy ra \(\left(\frac{1}{a}+\frac{1}{b}\right)\left(a+b\right)\ge4\RightarrowĐPCM\)\(\Rightarrow\frac{1}{a+b}\le\frac{1}{4}\left(\frac{1}{a}+\frac{1}{b}\right)\left(1\right)\)

a/Áp dụng (1) có

\(\frac{1}{a+b+2c}\le\frac{1}{4}\left(\frac{1}{a+c}+\frac{1}{b+c}\right)\left(2\right)\).Tương tự ta cũng có:

\(\frac{1}{b+c+2a}\le\frac{1}{4}\left(\frac{1}{a+b}+\frac{1}{a+c}\right)\left(3\right),\frac{1}{c+a+2b}\le\frac{1}{4}\left(\frac{1}{b+c}+\frac{1}{a+b}\right)\left(4\right)\)

Cộng (2),(3) và (4) có \(VT\le\frac{1}{4}.\left(6+6\right)=3\left(ĐPCM\right)\)

b/Áp dụng (1) có:

\(\frac{1}{3a+3b+2c}=\frac{1}{\left(a+b+2c\right)+2\left(a+b\right)}\le\frac{1}{4}\left(\frac{1}{a+b+2c}+\frac{1}{2\left(a+b\right)}\right)\left(5\right)\)

Tương tự có: \(\frac{1}{3a+2b+3c}\le\frac{1}{4}\left(\frac{1}{a+c+2b}+\frac{1}{2\left(a+c\right)}\right)\left(6\right)\)

\(\frac{1}{2a+3b+3c}\le\frac{1}{4}\left(\frac{1}{2a+b+c}+\frac{1}{2\left(b+c\right)}\right)\left(7\right)\)

Cộng (5),(6) và (7) có:

\(VT\le\frac{1}{4}\left(\frac{1}{a+b+2c}+\frac{1}{a+c+2b}+\frac{1}{2a+b+c}+\frac{1}{2}\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c}\right)\right)\le\frac{1}{4}.9=\frac{3}{2}\)