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

Answer

Range :`(-oo,2//e]`

Explanation

f(X) =`(4log_(e)x)/(x^(2))`
The domain is (0,oo)
`f(x) =(4log_(e)x)/(x^(2))=-oo`
and `underset(x rarr oo)lim(4log_(e)x)/(x^(2)) = underset(xrarroo)lim (4)/(x)(2x)=underset(xrarroo)lim (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) =0 when x =1
Derivative changes sign from +ve to -ve in neightbourhood of `x = e^(1//2)`
From the above discussion the graph of y = f(X) is as shown in the following figure

From the graph x =0 (y axis) and x =0 (y axis) are asymptotes
To the curve also of the function is `-oo,f(e^(1//2))or (-oo,2//e)`