Show that the maximum value of `(1/x)^x` is `e^(1//e)` .

Options

A. e
B. `e^( e)`
C. `e^(1//e)`
D. `(1/e)^(1//e)`

Answer

C

Explanation

Let `y=(1/x)^(x)`
`rArr logy=x.log1/x`
`therefore 1/y(dy)/(dx)=x.1/(1/x). (-1/x^(2))+ log 1/x.1`
`=-1+log1/x`
`therefore (dy)/(dx)= (log 1/x-1).(1/x)^(x)`
Now, `(dy)/(dx)=0`
`rArr log1/x=1=loge`
`rArr 1/x=e`
`x=1/e`
Hence, the maximum value of `f(1/e)=(e)^(1//e)`