Question

Cho a, b, c > 0 thỏa mãn : \(ab+bc+ca=2015\) .

CMR: A= \(\sqrt{2015+a^2}+\sqrt{2015+b^2}+\sqrt{2015+c^2}\le2\left(a+b+c\right)\)

Answer 1

- Ta có: \(A=\sqrt{2015+a^2}+\sqrt{2015+b^2}+\sqrt{2015+c^2}\)

\(=\sqrt{a^2+ab+bc+ca}+\sqrt{b^2+ab+bc+ca}+\sqrt{c^2+ab+bc+ca}\)

\(=\sqrt{\left(a+b\right)\left(c+a\right)}+\sqrt{\left(b+c\right)\left(a+b\right)}+\sqrt{\left(c+a\right)\left(b+c\right)}\)

\(\le\dfrac{\left(a+b\right)+\left(c+a\right)}{2}+\dfrac{\left(b+c\right)+\left(a+b\right)}{2}+\dfrac{\left(c+a\right)+\left(b+c\right)}{2}\)

\(=2\left(a+b+c\right)\left(đpcm\right)\)

- Dấu "=" xảy ra khi \(a=b=c=\sqrt{\dfrac{2015}{3}}\)