Explanation

(a)
Since, $AM \geq GM$
$\therefore\frac{a^{2} + b^{2}}{2} \geq \sqrt{a^{2}b^{2}} = ab,\frac{b^{2} + c^{2}}{2} \geq bc$
and $\frac{c^{2} + a^{2}}{2} \geq ca$
On adding, we get
$a^{2} + b^{2} + c^{2} \geq ab + bc + ca$
$\Rightarrow (a)\$holds
Next, $\frac{b + c}{2} \geq \sqrt{bc},\ \ \frac{c + a}{2} \geq \sqrt{ca}$,$\ \frac{a + b}{2} \geq \sqrt{ab}$
$\Rightarrow \left( \frac{b + c}{2} \right)\left( \frac{c + a}{2} \right)\left( \frac{a + b}{2} \right) \geq \sqrt{a^{2}b^{2}c^{2}}$
$\Rightarrow (b + c)(c + a)(a + b) \geq 8abc$
$\Rightarrow (b)$ does not hold
Again, $\frac{1}{3}\left( \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \right) \geq \left( \frac{a}{b} \bullet \frac{b}{c} \bullet \frac{c}{a} \right)^{1/3}$
$\Rightarrow \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq 3$
$\Rightarrow (c$)does not hold
Again, $\frac{a^{3} + b^{3} + c^{3}}{3} \geq \left( a^{3}b^{3}c^{3} \right)^{1/3}$
$\Rightarrow a^{3} + b^{3} + c^{3} \geq 3\ abc$
$\Rightarrow (d)$ does not hold