The least value of α ∈ R for which 4 α x2+(1/x) ≥ 1, for all x0, is -

Question

The least value of α ∈ R for which 4 α x2+(1/x) ≥ 1, for all x>0, is - (A) (1/64) (B) (1/32) (C) (1/27) (D) (1/25)

Tardigrade Admin · Accepted Answer

(4 α x2+(1/2 x)+(1/2 x)/3) ≥((4 α x2/4 x2))1 / 3 ( where α>0 obviously) 4 α x2+(1/x) ≥ 3 α1 / 3 ∴ 3 α1 / 3 ≥ 1 α ≥ (1/27)

Q. The least value of $α \in R$ for which $4 α x^{2} + \frac{1}{x} \geq 1$ , for all $x > 0$ , is -