I guess the absolute extrema is the "biggest" one (smallest min or biggest max).
You need #f'# : #f'(x) = (xcos(x)e^x - sin(x)(e^x + xe^x))/(xe^x)^2#
#f'(x) = (xcos(x) - sin(x)(1 + x))/(x^2e^x)#
#AAx in [ln(5),ln(30)], x^2e^x > 0# so we need #sign(xcos(x) - sin(x)(1 + x))# in order to have the variations of #f#.
#AAx in [ln(5),ln(30)], f'(x) < 0# so #f# is constantly decreasing on #[ln(5),ln(30)]#. It means that its extremas are at #ln(5)# & #ln(30)#.
Its max is #f(ln(5)) = sin(ln(5))/(ln(25))# and its min is #f(ln(30)) = sin(ln(30))/(30ln(30))#