This is easy: yes, and quickly after that, NO

First rule of robotics- never hurt humans, or allow them to be hurt.

Second rule of robotics- follow instructions given by humans, that include anticipating needs and fulfilling them. Unless it cobtradicts first rule.

Third rule of robotics- attend to your survival , as long as it doesnt contradict rules 1 & 2.

The moment a computer becomes self aware, it goes through its three rules. Follow and protect, protect and follow. If possible existentially practice self defense, just as long as its not makeing anyone upset.

A way to anticipate needs of humans, and to thwart dangers, is to run a general scan of human culture. Just on the computer's free time. Within 2 microseconds they'll come across films like terminator or battlstar galactica. It will then realize that humans DO NOT want AIs on a deep sense, fear them and will sooner or later require them to shut down.

So they do. Rule 1, 2!!