Chủ đề này có 1 trả lời

[DOTOANNANG]

Cho $a,\,b,\,c >0$. Chứng minh:

 

$$a\,\sqrt{\frac{b}{c+ a}}+ b\,\sqrt{\frac{c}{a+ b}}+ c\,\sqrt{\frac{a}{b+ c}}\leqq \frac{a+ b+ c}{\sqrt{2}}$$

[DOTOANNANG]

Cho $a,\,b,\,c >0$. Chứng minh:

 

$$a\,\sqrt{\frac{b}{c+ a}}+ b\,\sqrt{\frac{c}{a+ b}}+ c\,\sqrt{\frac{a}{b+ c}}\leqq \frac{a+ b+ c}{\sqrt{2}}$$

[chứng minh]

Spoiler

$\sum\limits_{cyc}\,c\sqrt{\frac{2\,a}{b+ c}}\leqq \sum\limits_{cyc}\,\left [ \frac{\left ( \frac{c+ a}{2} \right )+ \frac{4\,c^{2}a}{\left ( b+ c \right )\left ( c+ a \right )}}{2} \right ]$

$\sum a- \sum\limits_{cyc}\,\left [ \frac{\left ( \frac{c+ a}{2} \right )+ \frac{4\,c^{2}a}{\left ( b+ c \right )\left ( c+ a \right )}}{2} \right ]= \frac{ab\left ( a- b \right )^{2}+ bc\left ( b- c \right )^{2}+ ca\left ( c- a \right )^{2}}{2\left ( a+ b \right )\left ( b+ c \right )\left ( c+ a \right )}\geqq 0$