Question

If $a,b,c$ are three positive numbers then

• A. $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}<3$
• B. $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge 9$
• C. $a+b+c>3(abc)$
• D. $a^{2}b+b^{2}c+c^{2}a\ge 9abc$

Answer 1

Answer: (B)

(1) $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge {\left(\frac{a}{b}\cdot \frac{b}{c}\cdot \frac{c}{a}\right)}^{1/3}$
$\Rightarrow \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge 3$
(2) $AM\ge HM\Rightarrow \frac{a+b+c}{3}\ge \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$
(3) $\frac{a+b+c}{3}\ge (abc{)}^{1/3}$
(4) $\frac{a^{2}b+b^{2}c+c^{2}a}{3}\ge {\left(a^3{b}^3{c}^3\right)}^{1/3}$
$\Rightarrow a^{2}b+b^{2}c+c^{2}a\ge 3abc$