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If $a, b, c$ are the positive integers, then $(a+b)(b+c)(c+a)$ is
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Verified Answer
The correct answer is:
$\geq 8 a b c$
We know that,
$A M \geq G M$
$\Rightarrow \frac{a+b}{2} \geq \sqrt{a b}$\ldots(i)
Similarly,
$\frac{b+c}{2} \geq \sqrt{b c}$\ldots(ii)
$\frac{c+a}{2} \geq \sqrt{c a}$\ldots(iii)
Multiplying these inequalities, we will get
$\frac{(a+b)(b+c)(c+a)}{2 \times 2 \times 2} \geq \sqrt{a^2 b^2 c^2}$
$\Rightarrow(a+b)(b+c)(c+a) \geq 8 a b c$
Hence, $\geq$ 8abc is the correct answer.
$A M \geq G M$
$\Rightarrow \frac{a+b}{2} \geq \sqrt{a b}$\ldots(i)
Similarly,
$\frac{b+c}{2} \geq \sqrt{b c}$\ldots(ii)
$\frac{c+a}{2} \geq \sqrt{c a}$\ldots(iii)
Multiplying these inequalities, we will get
$\frac{(a+b)(b+c)(c+a)}{2 \times 2 \times 2} \geq \sqrt{a^2 b^2 c^2}$
$\Rightarrow(a+b)(b+c)(c+a) \geq 8 a b c$
Hence, $\geq$ 8abc is the correct answer.
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