Cho \(a,b>0:a+b\le2\).Tìm max: P=\(\sqrt{a\left(b+3\right)}+\sqrt{b\left(a+3\right)}\)

Cho \(a,b>0:\dfrac{a}{1+a}+\dfrac{b^3}{b+1}=1\). Tìm max: P=\(ab^3\)

Cho \(a,b,c>0\). Chứng minh:

\(\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}< \sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}\)

Cho \(a,b>0:ab+1\le b\). Chứng minh:

\(\left(a+\dfrac{1}{a^2}\right)+\left(b^2+\dfrac{1}{b}\right)\ge9\)

Cho a, b: \(2a^2+5b^2+2ab=1\)

Chứng minh: \(-\dfrac{1}{\sqrt{3}}\le\dfrac{a-b}{a+2b+2}\le\dfrac{1}{\sqrt{3}}\)

a, b > 0. Chứng minh:\(\dfrac{1}{a^3}+\dfrac{a^3}{b^3}+b^3\ge\dfrac{1}{a}+\dfrac{a}{b}+b\)

Cho a, b,c : abc = 1. Chứng minh:

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

Cho \(x,y,z\in[0,1]\). Chứng minh:

\(\dfrac{x}{yz+1}+\dfrac{y}{xz+1}+\dfrac{z}{xy+1}< 2\)

a, Cho x, y, z > 0 \(\in[0,1]\). Chứng minh:

\(\dfrac{x}{yz+1}+\dfrac{y}{xz+1}+\dfrac{z}{xy+1}< 2\)

b, x, y, z  > 0 : xyz = 1. Chứng minh:

\(\dfrac{1}{x^2+2y+3}+\dfrac{1}{y^2+2z^2+3}+\dfrac{1}{z^2+2x^2+3}\le2\)

a, b, c là độ dài 3 cạnh tam giác. Chứng minh:

a, 1 < \(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}< 2\)

b, 1 < \(\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{a+c}}+\sqrt{\dfrac{c}{a+b}}\)