Let a, b, c be positive real numbers, such that b x2+ (√(a+c)2+4 b2) x+(a+c) ≥ 0, ∀ x ∈ R, then a, b, c are in:

Question

Let a, b, c be positive real numbers, such that b x2+ (√(a+c)2+4 b2) x+(a+c) ≥ 0, ∀ x ∈ R, then a, b, c are in: (A) G.P. (B) A.P. (C) H.P. (D) None of these

Tardigrade Admin · Accepted Answer

(a+c)2+4 b2-4 b(a+c) ≤ 0 ⇒ (a-2 b+c)2 ≤ 0 ⇒ a-2 b+c=0 ⇒ 2 b=a+c ⇒ a, b, c are in A.P.

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

G.P.

A.P.

H.P.

None of these

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

Q. Let a,b,c be positive real numbers, such that bx2+ ((a+c)2+4b2​)x+(a+c)≥0,∀x∈R, then a,b,c are in: