Verified Answer :

apologiabiology

take the deritivive
remember
the deritivive of f(x)/g(x)=(f'(x)g(x)-g'(x)f(x))/(g(x)^2)
so
deritiveive is ln(x)/x is
remember that derivitive of lnx is 1/x
so

(1/x*x-1lnx)/(x^2)=(1-ln(x))/(x^2)
the max occurs where the value is 0
(1-ln(x))/(x^2)=0
times x^2 both sides
1-lnx=0
add lnx both sides
1=lnx
e^1=x
e=x
see if dats a max or min
at e/2, the slope is positive
at 3e/2, the slope is negative
changes from positive to negative at x=e

that means it's a max

max at x=e
I realize I didn't find the max point, so

sub back
ln(x)/x
ln(e)/e
1/e
the value of the max would be 1/e occuring where x=e

4th option is answer (1/e) because that is the value of the maximum (which happens at x=e)

The absolute maximum of the function ln(x)/x is (e, 1/e)
To find the minimums/maximums of a function, you must find the critical points of the function.
f(x) = ㏑(x)÷x
f'(x) = ((1/x · x) - ㏑(x)) ÷ x² =
(1 - ㏑(x)) ÷ x² = 0
1 - ㏑(x) = 0
㏑(x) = 1
x = e 
We have determined that our only critical point is e, which means that that is the absolute maximum of the function. 
f(e) = ㏑(e) ÷ e = 1/e
The relative (absolute) maximum of the function ln(x)/x is 3)


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