Hi,
this should work:
NMaximize[Sqrt[Abs[x]]*BesselJ[1, x], 0 < x < 10, x]
and likewise
NMaximize[Sort[x]*Abs[BesselJ[2, x] - BesselJ[0, x]], 0 < x < 10, x]
If you want to work with interval, this should help
NMaximize[
Sqrt[Abs[x]]*BesselJ[1, x], {x} \[Element] Interval[{0, 10}]]
If you want to increase the precision, this should do
NMaximize[Sqrt[Abs[x]]*BesselJ[1, x], {x} \[Element] Interval[{0, 10}], WorkingPrecision -> 30]
Cheers,
Marco
PS: Sorry posted nearly at the same time as the previous answer.