State True or False.
If p, q, r, s are positive real numbers, then $(p^{2} + p + 1) (q^{2} + q + 1) (r^{2} + r + 1) (s^{2} + s + 1) \geq 81 p q r s$ .

True

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False

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Solution

The correct option is A True
$p \in R^{+}$ $\Rightarrow \frac{p^{2} + p + 1}{3} \geq (p^{2} . p .1)^{\frac{1}{3}}$
since, $A P \geq G P$
$p^{2} + p + 1 \geq 3 p$
Similarly, writing the other three inequalities; by symmetry and multiplying;
$(p^{2} + p + 1) (q^{2} + q + 1) (r^{2} + r + 1) (s^{2} + s + 1) \geq 81 p q r s$

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