Question

Which of the following hold good?

• A. $a^4+b^4+c^4>abc(a+b+c)$
• B. $a^5+b^5+c^5+d^5>abcd(a+b+c+d)$
• C. $a^5+b^5+c^5>abc(ab+bc+ca)$
• D. $\frac{a^8+b^8+c^8}{a^3{b}^3{c}^3}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$
• E. $\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b}>a+b+c$

Answer 1

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

The correct options are
A $a^4+b^4+c^4>abc(a+b+c)$
B $a^5+b^5+c^5+d^5>abcd(a+b+c+d)$
C $a^5+b^5+c^5>abc(ab+bc+ca)$
D $\frac{a^8+b^8+c^8}{a^3{b}^3{c}^3}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

(a), (b), (c), (d)
(a) $\frac{a^4+b^4+c^4}{3}>{\left(\frac{a+b+c}{3}\right)}^4$
or $\left(\frac{a+b+c}{3}\right){\left(\frac{a+b+c}{3}\right)}^3>\frac{a+b+c}{3}\left[\right(abc{)}^{\frac{1}{3}}{]}^3$;
$\because A.M.>G.M.$
or $>\left(\frac{a+b+c}{3}\right)$
$\therefore a^4+b^4+c^4>abc(a+b+c)$

(b) As above

(c) $\left(\frac{a^5+b^5+c^5}{3}\right)>{\left(\frac{a+b+c}{3}\right)}^5$
or $>{\left(\frac{a+b+c}{3}\right)}^3{\left(\frac{a+b+c}{3}\right)}^2$
or $>\left[\right(abc{)}^{\frac{1}{3}}{]}^3\left(\frac{a^2+b^2+c^2+2ab+2bc+2ca}{9}\right)$
But we know that $a^2+b^2+c^2>ab+bc+ca$ by result of two by two rule.
$\therefore \frac{a^5+b^5+c^5}{3}>abc\frac{3(ab+bc+ca)}{9}$ etc.

(d) $a^8+b^8+c^8>a^2{b}^2{c}^2(bc+ca+ab)$
Now $\frac{a^8+b^8+c^8}{3}>{\left(\frac{a+b+c}{3}\right)}^8$
or $>{\left(\frac{a+b+c}{3}\right)}^6{\left(\frac{a+b+c}{3}\right)}^2>\left[\right(abc{)}^{\frac{1}{3}}{]}^6\left[\frac{a^2+b^2+c^2+2ab+2bc+2ca}{9}\right]$
$\because A.M.>G.M$
But by two by two rule
$a^2+b^2+c^2>ab+bc+ca$
$\therefore \frac{a^8+b^8+c^8}{3}>a^2{b}^2{c}^2\frac{(3ab+3bc+3ca)}{9}$
$\therefore a^8+b^8+c^8>a^2{b}^2{c}^2(ab+bc+ca)$
or $\frac{a^8{b}^8{c}^8}{a^3{b}^3{c}^3}>\frac{ab+bc+ca}{abc}$ or $>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

(e) $\frac{b^2+c^2}{2}>{\left(\frac{b+c}{2}\right)}^2\therefore \frac{b^2+c^2}{b+c}>\frac{b+c}{2}$
Write similar inequalities and add.