Question

The least natural number a for which $x+a{x}^{-2}>2forallxϵR^{+}$

• A. 1
• B. 2
• C. 5
• D. 9

Answer 1

The correct option is B 2

x + $\frac{a}{x^2}>2\forall xϵR^{+}{x}^3-2x^2+a>0$
f(x) = $x^3-2x^2+a$
f'(x) = $3x^2-4x=x(3x-4)$
f$\left(\frac{4}{3}\right)>0{\left(\frac{4}{3}\right)}^3-2{\left(\frac{4}{3}\right)}^2+a>0$
$\frac{64}{27}-\frac{32}{9}+a>0$
$\frac{64-96}{27}+a>0$
$a>\frac{32}{27}$ least natural number is 2.