Question

Which one of the following hold good?

• A. $b^2{c}^2+c^2{a}^2+a^2{b}^2\ge abc(a+b+c)$
• B. $\frac{bc}{a}+\frac{ca}{b}+\frac{ab}{c}\ge a+b+c$
• C. $\frac{bc}{a^3}+\frac{ca}{b^3}+\frac{ab}{c^3}\ge \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$
• D. $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{1}{\sqrt{\left(bc\right)}}+\frac{1}{\sqrt{\left(ca\right)}}+\frac{1}{\sqrt{\left(ab\right)}}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $b^2{c}^2+c^2{a}^2+a^2{b}^2\ge abc(a+b+c)$
B $\frac{bc}{a}+\frac{ca}{b}+\frac{ab}{c}\ge a+b+c$
C $\frac{bc}{a^3}+\frac{ca}{b^3}+\frac{ab}{c^3}\ge \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$
D $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{1}{\sqrt{\left(bc\right)}}+\frac{1}{\sqrt{\left(ca\right)}}+\frac{1}{\sqrt{\left(ab\right)}}$

(a), (b), (c), (d)
(a) $b^2{c}^2+c^2{a}^2>2(a^2{b}^2{c}^4{)}^{\frac{1}{2}}or>2ab{c}^2$
Similarly $c^2{a}^2+a^2{b}^2>2bc{a}^2$
$a^2{b}^2+b^2{c}^2>2ca{b}^2$
Add all three inequalities

(b) $\frac{bc}{a}+\frac{ca}{b}\ge 2{(\frac{bc}{a}.\frac{ca}{b})}^{\frac{1}{2}}or\ge 2c$ etc.

(c) $\frac{bc}{a^3}+\frac{ca}{b^3}>2{\left(\frac{ba{c}^2}{a^3{b}^3}\right)}^{\frac{1}{2}}or>\frac{2c}{ab}$
Similarly writing other inequalities and adding, we get
$\frac{bc}{a^3}+\frac{ca}{b^3}+\frac{ab}{c^3}>\frac{c}{ab}+\frac{b}{ca}+\frac{a}{bc}\dots \dots \left(A\right)$
Again $\frac{c}{ab}+\frac{b}{ca}>2{\left[\frac{bc}{bc.a^2}\right]}^{\frac{1}{2}}=2.\frac{1}{a}$
Similarly writing other inequalities and adding, we get
$\frac{c}{ab}+\frac{b}{ca}+\frac{a}{bc}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\dots \dots \left(B\right)$
$\therefore \frac{bc}{a^3}+\frac{ca}{b^3}+\frac{ab}{c^3}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ from (A) and (B)

(d) $\frac{1}{b}+\frac{1}{c}>2{[\frac{1}{b}.\frac{1}{c}]}^{\frac{1}{2}}or>2\frac{1}{\sqrt{\left(bc\right)}}$
and so on