MCQ
Let ${a_1},{a_2},{a_3}$ be any positive real numbers, then which of the following statement is not true
  • A
    $3{a_1}{a_2}{a_3} \le a_1^3 + a_2^3 + a_3^3$
  • B
    $\frac{{{a_1}}}{{{a_2}}} + \frac{{{a_2}}}{{{a_3}}} + \frac{{{a_3}}}{{{a_1}}} \ge 3$
  • C
    $({a_1} + {a_2} + {a_3})\,\left( {\frac{1}{{{a_1}}} + \frac{1}{{{a_2}}} + \frac{1}{{{a_3}}}} \right) \ge 9$
  • $({a_1} + {a_2} + {a_3})\,{\left( {\frac{1}{{{a_1}}} + \frac{1}{{{a_2}}} + \frac{1}{{{a_3}}}} \right)^3} \le 27$

Answer

Correct option: D.
$({a_1} + {a_2} + {a_3})\,{\left( {\frac{1}{{{a_1}}} + \frac{1}{{{a_2}}} + \frac{1}{{{a_3}}}} \right)^3} \le 27$
d
(d) $G.M$ $ \ge $ $H.M.$

==> ${({a_1}.{a_2}.{a_3})^{1/3}} \ge \frac{3}{{(1/{a_1} + 1/{a_2} + 1/{a_3})}}$

==> $({a_1}.\,{a_2}.{a_3})\, \ge \,\frac{{27}}{{{{\left( {1/{a_1} + 1/{a_2} + 1/{a_3}} \right)}^3}}}$

$({a_1}.\,{a_2}.{a_3})\,{\left( {\frac{1}{{{a_1}}} + \frac{1}{{{a_2}}} + \frac{1}{{{a_3}}}} \right)^3}$$ \ge 27$.

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