The maximum value of `f(x)=(logx)/(x)(x ne 0, x ne1)` is

Options

A. e
B. `(1)/(e)`
C. `e^(2)`
D. `(1)/(e^(2))`

Answer

B

Explanation

Given, `f(x)=(logx)/(x)` . . . (i)
On differentiating both sides w.r.t.x, we get
`f'(x)=(x*(1)/(x)-logx*1)/(x^(2))=(1-logx)/(x^(2))`
For maximum or minimum value, put f'(x)=0
`(1-logx)/(x^(2))=0implieslogx=1impliesx=e`
On substituting x=e in Eq. (i), we get
`f(e)=(loge)/(e)=(1)/(2) lt 0`
`therefore`Maximum value of `f(x)` is `(1)/(e)` at x=e.