Giai cho em bai BDT kho nay voi

Q

quanghero100

tronganh11 said:
a/b+b/c+c/a <b/a+c/b+a/c Voi a>b>c>0
Voi dieu kien nhu tren hay giups em voi
Bấm để xem đầy đủ nội dung ...
với điều kiện a>b>c>0
Ta có:
[TEX] \frac{a}{b}+\frac{b}{c}+\frac{c}{a} < \frac{b}{a}+\frac{c}{b}+\frac{a}{c}\\ \Leftrightarrow \frac{a^2c}{abc}+ \frac{b^2a}{abc}+ \frac{c^2b}{abc}<\frac{b^2c}{abc}+\frac{c^2a}{abc}+\frac{a^2b}{abc}\\ \Leftrightarrow a^2c+b^2a+c^2b-b^2c-c^2a-a^2b<0\\ \Leftrightarrow ac(a-c)+ab(b-a)+cb(c-b)<0\\ \Leftrightarrow a(ca-c^2+b^2-ab)-cb(b-c)<0\\ \Leftrightarrow a(b-c)(b+c-a)-cb(b-c)<0\\ \Leftrightarrow (b-c)[a(b+c-a)-cb]<0\\ \Leftrightarrow (b-c)[a(b-a)-c(b-a)]<0\\ \Leftrightarrow (b-c)(a-c)(b-a)<0 [/TEX](đúng) vì (b-c)>0; (a-c)>0; (b-a)<0
 
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