Am I required to replace my fire extinguishers periodically?

That depends on the type of extinguishers that you own. Some extinguishers are not serviceable or rechargeable (i.e. they are Single-Use). Some extinguishers are more cost effective to replace with a new one, than to service or maintain them.

Below is a list of extinguisher types that are considered obsolete and the state fire marshal requires that they be removed from service.
577.2. Obsolete Fire Extinguishers.

The following types of fire extinguishers are considered obsolete and shall be removed from service:
(1) Soda acid types
(2) Chemical foam (excluding AFFF and FFFP)
(3) Vaporizing liquid (e.g., carbon tetrachloride)
(4) Cartridge-operated water
(5) Cartridge-operated loaded stream
(6) Copper or brass shell fire extinguishers (excluding pump tanks) joined by soft solder or rivets
(7) Stored pressure water extinguishers with fiberglass shells
(8) Solid charge-type AFFF extinguishers (paper cartridge)
(9) Pressurized water fire extinguishers manufactured prior to 1971
(10) Any extinguisher that needs to be inverted to operate
(11) Any stored pressure extinguisher manufactured prior to 1955
(12) Any extinguishers with 4B, 6B, 8B, 12B, and 16B fire ratings
(13) Dry chemical stored pressure extinguishers, other than wheeled type, manufactured prior to
October 1984 shall be removed from service at the next 6-year maintenance interval or the next
hydrostatic test interval, whichever comes first.

Note: Authority cited: Section 13160, Health and Safety Code. Reference: Section 13160, Health and Safety Code.