Do you mean "maximize AppletViewer when it runs" or "maximize the window in which the applet runs"? (Remember, an applet runs in a Web browser, and not in a separate viewer window like a Java application would.)
If the second, in an application you can get the resolution of the monitor, then set the size to the x and y size. In an applet, I don't know if you can do that. The Applet class has reSize and setSize methods, and it inherits the getMaximumSize and getMinimumSize methods from class Container and other getter/setters from class Component, so this may be a good starting point for experimentation.
Give a man a fish, he'll eat for one day. Teach a man to fish, he'll drink all your beer.
Cheers, Jeff (SCJP 1.4, SCJD in progress, if you can call that progress...)