Question

If $a,b,c$ are three positive numbers,then the minimum value of the expression $\frac{ab(a+b)+bc(b+c)+ca(c+a)}{bca}$

• A. $3$
• B. $4$
• C. $6$
• D. $1$

Answer 1

Answer: (C)

$6$

Given expression
$\frac{(a+b)}{c}+\frac{(b+c)}{a}+\frac{(c+a)}{b}$
or $\frac{a}{c}+\frac{b}{c}+\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}$
Using $AM\ge GM$
$\frac{\frac{a}{c}+\frac{b}{c}+\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}}{6}\ge \sqrt{\frac{a}{c}\frac{b}{c}\frac{b}{a}\frac{c}{a}\frac{c}{b}\frac{a}{b}}=6$
Hence, option 'C' is correct.