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)
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)
