CMR: \(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}>=\sqrt{\dfrac{a^2+b^2}{2}}+\sqrt{\dfrac{b^2+c^2}{2}}+\sqrt{\dfrac{c^2+a^2}{2}}\left(a,b,c>0\right)\)
Cho a,b,c>0 thỏa mãn ab+bc+ca=1. CMR:
\(\left(\dfrac{a}{\sqrt{a^2+1}}+\dfrac{b}{\sqrt{b^2+1}}+\dfrac{c}{\sqrt{c^2+1}}\right)^3\le\dfrac{3}{2}\left(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\right)\)
Cho các số thực dương a+b+c=\(\sqrt{a}+\sqrt{b}+\sqrt{c}=2\\\).CMR
\(\dfrac{\sqrt{a}}{1+a}+\dfrac{\sqrt{b}}{1+b}+\dfrac{\sqrt{c}}{1+c}=\dfrac{2}{\sqrt{\left(1+a\right)\left(1+b\right)\left(1+c\right)}}\)
cho a,b,c>0 thỏa mãn \(a^2+b^2+c^2=1\).CMR
\(\dfrac{\sqrt{ab+2c^2}}{\sqrt{1+ab-c^2}}+\dfrac{\sqrt{bc+2a^2}}{\sqrt{1+bc-a^2}}+\dfrac{\sqrt{ca+2b^2}}{\sqrt{1+ca-b^2}}\ge2+ab+bc+ca\)
Cho a,b,c là các số thực dương thỏa mãn \(\sqrt{a}+\sqrt{b}+\sqrt{c}=2\). Chứng minh rằng:\(\dfrac{a+b}{\sqrt{a}+\sqrt{b}}+\dfrac{b+c}{\sqrt{b}+\sqrt{c}}+\dfrac{c+a}{\sqrt{c}+\sqrt{a}}\le4\left(\dfrac{\left(\sqrt{a}-1\right)^2}{\sqrt{b}}+\dfrac{\left(\sqrt{b}-1\right)^2}{\sqrt{c}}+\dfrac{\left(\sqrt{c}-1\right)^2}{\sqrt{a}}\right)\)
Cho a,b,c là các số thực dương. Chứng minh rằng :
\(\dfrac{a}{\sqrt{b^2+\dfrac{bc}{4}+c^2}}+\dfrac{b}{\sqrt{c^2+\dfrac{ca}{4}+a^2}}+\dfrac{c}{\sqrt{a^2+\dfrac{ba}{4}+b^2}}\ge2\)
a+b+c=\(\sqrt{a}+\sqrt{b}+\sqrt{b}=2\)
Cmr: \(\dfrac{\sqrt{a}}{1+a}+\dfrac{\sqrt{b}}{1+b}+\dfrac{\sqrt{c}}{1+c}=\dfrac{2}{\sqrt{\left(1+a\right)\left(1+b\right)\left(1+c\right)}}\)
Cho a,b,c > 0 thỏa a+b+c=abc. Tìm GTLN của BT :
\(\dfrac{a}{\sqrt{bc\left(1+a^2\right)}}+\dfrac{b}{\sqrt{ac\left(1+b^2\right)}}+\dfrac{c}{\sqrt{ab\left(1+c^2\right)}}\)
C=(\(B=\left(\dfrac{1}{\sqrt{x-1}}-\dfrac{1}{\sqrt{a}}\right):\left(\dfrac{\sqrt{x}+1}{\sqrt{a}-2}-\dfrac{\sqrt{a}+2}{\sqrt{a}-2}\right)\) a Tìm đkxd của B
b rút gọn B
c tìm a sao cho B \(\le\)\(\dfrac{1}{3}\)