Draw and graph of `f(x) = (4log_(e)x)/(x^(2))`. Also find the range.

Explanation

`f(x) = (4log_(e)x)/(x^(2))`
The domain is `(0, infty)`.
`f^(')(x) =4(1/x(x^(2)-2xlog_(e)x))/(x^(4))=4(1-2log_(e)x)/(x^(3))`
`f^(')(x) =0 rArr 1-2log_(e)x=0` or `x = e^(1//2)`
`lim_(x to 0^(+))(4log_(e)x)/(x^(2))=-infty`
and `lim_(xtoinfty)(4log_(e)x)/(x^(2)) = lim_(xto infty)(4/x)/(2x)=lim_(xtoinfty)2/x^(2)=0`
`f(x) lt 0` for `0 lt x lt 1` and `f(x) gt 0` for `x gt 1`
`f(x) lt 0` for `0 lt x lt ` and `f(x) gt0` for `x gt 1`
`f(x)=0` ,When `x=1`
Derivative changes sign from `+ve` to `-ve` in the neighbourhood of `x=e^(1//2)`
From the above discussion, the graph of `y=f(x)` is as shown in the following figure.

Form the graph, `x=0`(y-axis) and `x=0` (y-axis) are asymptotes to the curve. Also the range of the function is `(-infty, fe^(1//2))` or `(-infty, 2//e)`