Question

Let $a, b, c$ be positive real numbers, such that $b x^{2}+$ $\left(\sqrt{(a+c)^{2}+4 b^{2}}\right) x+(a+c) \geq 0, \forall x \in R$, then $a, b, c$ are in:

• A. G.P.
• B. A.P.
• C. H.P.
• D. None of these

Answer 1

Answer: (B)

$(a+c)^{2}+4 b^{2}-4 b(a+c) \leq 0$
$\Rightarrow (a-2 b+c)^{2} \leq 0$
$\Rightarrow a-2 b+c=0$
$\Rightarrow 2 b=a+c$
$\Rightarrow a, b, c$ are in A.P.