a) Biến đổi vế phải, ta có :\(\frac{-3x\left(x-y\right)}{y^2-x^2}=\frac{3x\left(x-y\right)}{x^2-y^2}=\frac{3x\left(x-y\right)}{\left(x-y\right)\left(x+y\right)}=\frac{3x}{x+y}\) = vế trái \(\Rightarrowđpcm\)
c)Biến đổi vế phải ta có: \(\frac{3a\left(x+y\right)^2}{9a^2\left(x+y\right)}=\frac{x+y}{3a}=vt\Rightarrowđpcm\)

Dễ thấy: \(a^2;b^2;c^2\ge0\forall a;b;c\) mà \(a;b;c\ne0\) nên chỉ có \(a,b,c>0\)

Áp dụng BĐT AM-GM ta có: 

\(a^2+\frac{1}{a^2}\ge2\sqrt{a^2\cdot\frac{1}{a^2}}=2\sqrt{1}=2\)

\(b^2+\frac{1}{b^2}\ge2\sqrt{b^2\cdot\frac{1}{b^2}}=2\sqrt{1}=2\)

\(c^2+\frac{1}{c^2}\ge2\sqrt{c^2\cdot\frac{1}{c^2}}=2\sqrt{1}=2\)

Nhân theo vế 3 BĐT trên ta có: 

\(\left(a^2+\frac{1}{a^2}\right)\left(b^2+\frac{1}{b^2}\right)\left(c^2+\frac{1}{c^2}\right)\ge2\cdot2\cdot2=8\)

Đẳng thức xảy ra khi \(a=b=c\)

9a² + 4b² = 13ab => (3a)² + 2.3a.2b + (2b)² = 25ab => (3a+2b)² = 25ab => 3a + 2b  = 5\(\sqrt{ab}\) (do 3a ; 2b > 0)

9a² + 4b² = 13ab => (3a)² - 2.3a.2b + (2b)² = ab => (3a- 2b)² = ab => 3a - 2b  = \(\sqrt{ab}\)  (ví 3a > 2b > 0)

A = \(\frac{ab}{\left(3a-2b\right)\left(3a+2b\right)}=\frac{ab}{\sqrt{ab}.5\sqrt{ab}}=\frac{1}{5}\)

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

\(\Rightarrow\left(a+b\right)^2+4\left(a+b\right)-12\ge0\)

\(\Leftrightarrow\left(a+b-2\right)\left(a+b+6\right)\ge0\)

\(\Leftrightarrow a+b-2\ge0\Rightarrow a+b\ge2\)

Ta có:

BĐT\(\Leftrightarrow\frac{3a^2+3a+3b^2+3b}{\left(b+1\right)\left(a+1\right)}+\frac{ab}{a+b}\le a^2+b^2+\frac{3}{2}\)

\(\Leftrightarrow\frac{3a^2+3b^2+3a+3b}{4}+\frac{ab}{a+b}\le a^2+b^2+\frac{3}{2}\)

\(\Leftrightarrow3a+3b+\frac{4ab}{a+b}\le a^2+b^2+6\)

\(\Leftrightarrow3a+3b+\frac{4ab}{a+b}\le a^2+b^2+2\left(ab+a+b\right)\)

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

Ta có:

\(VT=a+b+\frac{4ab}{a+b}\le a+b+\frac{\left(a+b\right)^2}{a+b}=2\left(a+b\right)\le\left(a+b\right)\left(a+b\right)=\left(a+b\right)^2\)

Dấu "=" xảy ra khi \(a=b=1\)

@Nguyễn Việt Lâm

1. D= 1/3 + 1/3.4 + 1/3.4.5 + 1/3.4.5....n < 1/2 + 1/3.4 + 1/4.5 + ...+ 1/ n.(n-1)

=> còn lại thì bạn có thể tự chứng minh

mk chả hiểu j