If f(x) = {(ln \: x)}^(2) at which of the following x values is the function both decreasing and concave up? x = 0 \\ x = e \\ x = {e}^(2) \\ x = {e}^(3) \\ x = (1)/(e) ​

Question

If


`f(x) = {(ln \: x)}^(2)`
at which of the following x values is the function both decreasing and concave up?

`x = 0 \\ x = e \\ x = {e}^(2) \\ x = {e}^(3) \\ x = (1)/(e)`
​

Answer 1

`x = (1)/(e)`

Explanation:

`f(x) = {(lnx)}^(2)`

`f1st(x) = (2 lnx )/(x) = 0`

`x < 1 \: \: f1 < 0 \: \: de creasing`

`f2nd(x) = \frac{2 - 2 lnx}{ {x}^(2) } = 0`

`x < e \: \: f2 > 0 \: \: concave \: up`

So x<1 is the condition which contains x=1/e