Question

The maximum value of `"1n x"/x` in (2, ∞) is ______.

पर्याय

• 1

• e

• 2/e

• 1/e

Answer 1

The maximum value of `"1n x"/x` in (2, ∞) is 1/e.

Explanation:

Let y = `"1n x"/x`

`dy/dx = (x. 1/x − "In x".1)/x^2`

= `1 − log x/x^2`

For maxima, put `dy/dx = 0`

`1 − "In − x"/x^2 = 0`

⇒ x = e

Now, `(d^2y)/dx^2 = (x^2(−1/x) − (1 − "1nx") 2x)/(x^2)^2`

At x = e we have `(d^2y)/dx^2 < 0`

∴ The maximum value at x = e is y = `1/e`