The least natural number `a` for which `x+a x^(-2)>2AAx in (0,oo)` is 1 (b) 2 (c) 5 (d) none of these

Options

A. 1
B. 2
C. 5
D. none of these

Answer

2

Explanation

Let f(X) =`x+ax^(-2)-2`
Let `f(x) =1 -2ax^(-3)=0 or x =(2a)^(1//3)`
Also `f(x) =6ax^(-4) or f (x)(2a)(1//3)gt0`
Thus x=`(2a)^(1//3)` is the point of minima for `x+ax^(-2)-2gt0` forall ax , we must have `f(2a)^(1//3)gt0`
or `(2a)^(1//3)+a(2a)^(-//3)-2gt0`
or `2a+a-2(2a)^(2//3)gt0`
or `27a^(3)gt32a^(2)`
or `agt32//27`
Hence the least value of a is 2