Question

cho a,b,c là các số thực dương thỏa mãn a+b+c=1

cm: căn(a+b)+căn(b+c)+căn(c+a)<= căn6

Answer 1

đặt \(A=\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\)

\(=>A^2=\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)^2\)

\(=>A^2\le\left[\left(\sqrt{a+b}\right)^2+\left(\sqrt{b+c}\right)^2+\left(\sqrt{c+a}\right)^2\right].3\)

\(=>A^2\le\left[2\left(a+b+c\right)\right]3=2.3=6\)

\(=>A\le\sqrt{6}\left(dpcm\right)\)

dấu"=" xảy ra<=>a=b=c=1/3

Answer 2

Ta có:\(\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)^2=\left(1.\sqrt{a+b}+1.\sqrt{b+c}+1.\sqrt{c+a}\right)^2\)

  \(\le\left(1+1+1\right)\left(a+b+b+c+c+a\right)=3.2=6\)

\(\Rightarrow\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\le\sqrt{6}\)

Dấu "=" xảy ra <=> a=b=c=1/3