If a, b, c are all +ive real numbers which one of the following holds good?

(b + c) (c + a) (a + b) > 8abc
or (1 - a) (1 - b) (1 - c) > 8abc if a + b + c = 1

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(a + b + c) (bc + ca + ab) > 9abc
or

(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) > 9

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(\frac{a}{e} + \frac{b}{f} + \frac{c}{g}) (\frac{e}{a} + \frac{f}{b} + \frac{g}{c}) > g

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x_{i} > 0, (i = 1, 2, \dots \dots n)

, then

(x_{1} + x_{2} + \dots \dots + x_{n}) (\frac{1}{x_{1})} + \frac{1}{x_{2}} + \dots \dots + \frac{1}{x_{n}}) \geq n^{2}

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a_{i}^{s}

are all +ive real numbers, then

(1 + a_{1} + a_{1}^{2}) (1 + a_{2} + a_{2}^{2}) \dots \dots (1 + a_{n} + a_{n}^{2}) \geq 3^{n} a_{1} a_{2} \dots a_{n}

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Solution

The correct options are
A (b + c) (c + a) (a + b) > 8abc
or (1 - a) (1 - b) (1 - c) > 8abc if a + b + c = 1
B (a + b + c) (bc + ca + ab) > 9abc
or $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) > 9$
C $(\frac{a}{e} + \frac{b}{f} + \frac{c}{g}) (\frac{e}{a} + \frac{f}{b} + \frac{g}{c}) > g$
D If $x_{i} > 0, (i = 1, 2, \dots \dots n)$ , then
$(x_{1} + x_{2} + \dots \dots + x_{n}) (\frac{1}{x_{1})} + \frac{1}{x_{2}} + \dots \dots + \frac{1}{x_{n}}) \geq n^{2}$
E If $a_{i}^{s}$ are all +ive real numbers, then
$(1 + a_{1} + a_{1}^{2}) (1 + a_{2} + a_{2}^{2}) \dots \dots (1 + a_{n} + a_{n}^{2}) \geq 3^{n} a_{1} a_{2} \dots a_{n}$
All the five (a), (b), (c), (d), (e) hold good.
Apply $A . M . \geq G . M .$ on each bracket and multiply
e.g. $(a + b + c) \geq 3 (a b c)^{\frac{1}{3}}$
$(b c + c a + a b) \geq 3 (b c . c a . a b)^{\frac{1}{3}} = 3 (a b c)^{\frac{2}{3}}$
Now multiply both.
Similarly all other parts hold good.

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