Cho a,b,c là số thực dương.cmr [tex]\frac{a+3c}{a+b}+\frac{a+3b}{a+c}+\frac{2a}{b+c}\geq 5[/tex]
Ta có: [tex]VT= \frac{a+c}{a+b}+\frac{a+b}{a+c}+2(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b})[/tex]
Áp dụng BĐT Cô-si, ta có: [tex]\frac{a+c}{a+b}+\frac{a+b}{a+c}\geqslant 2[/tex]
Áp dụng BĐT Svacxo, ta có: [tex]\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=\frac{a^2}{ab+ac}+\frac{b^2}{ab+bc}+\frac{c^2}{ac+bc}\geqslant \frac{(a+b+c)^2}{2(ab+bc+ac)}\geqslant \frac{3(ab+bc+ac)}{2(ab+bc+ac)}=\frac{3}{2}[/tex]
[tex]\Rightarrow VT= \frac{a+c}{a+b}+\frac{a+b}{a+c}+2(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b})\geqslant 2+2.\frac{3}{2}=5[/tex] (đccm)
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
