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Question

If $$a_{1}, a_{2}$$ and $$a_{3}$$ be any positive real numbers, then which of the following statement is true?


A
3a1a2a3a31+a32+a33
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B
a1a2+a2a3+a3a13
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C
(a1+a2+a3)(1a2+1a2+1a3)9
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D
(a1.a2.a3)(1a2+1a2+1a3)327
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Solution

The correct option is D $$(a_{1} . a_{2} . a_{3}) \left (\dfrac {1}{a_{2}} + \dfrac {1}{a_{2}} + \dfrac {1}{a_{3}}\right )^{3}\geq 27$$
We know that, $$GM \geq HM$$

$$\Rightarrow (a_{1}.a_{2}.a_{3})^{1/3} \geq \dfrac {3}{\left (\dfrac {1}{a_{1}} + \dfrac {1}{a_{2}} + \dfrac {1}{a_{3}}\right )}$$

$$\Rightarrow (a_{1}.a_{2}.a_{3}) \geq \dfrac {27}{\left (\dfrac {1}{a_{1}} + \dfrac {1}{a_{2}} + \dfrac {1}{a_{3}}\right )^{3}}$$

$$\Rightarrow (a_{1}.a_{2}.a_{3})  \left (\dfrac {1}{a_{1}} + \dfrac {1}{a_{2}} + \dfrac {1}{a_{3}}\right )^{3} \geq 27$$

Hence, option D is correct.

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