Question

The minimum value of `1/x log x` in the interval `[2, oo]` is

Options

• `1/2 log 2`

• `1/2 e`

• `1/e`

• minimum value does not exist

Answer 1

minimum value does not exist

Explanation:

Let `f(x) = logx/x`

`f"'"(x) = (1 - log x)/x^2` = 0 ⇒ `x = e`

`f"''"(x) = (-x - 2x(1 - log x))/x^4`

Put x = e

`f"''"(e) = (-e)/e^4 < 0`

∴ `f(x)` is maximum at `x =  e` and minimum value does not eixst.