A deterministic FA for $0^*1^*$ is required
To construct a DFA, note the following:


An empty string or any string containing only $0$s is accepted, so
any $0^*$ is accepted.
Zero or more $0$s followed by one or more $1$s is accepted, so any
$0^*11^*$ is accepted.
Zero or more $0$s followed by one or more $1$s followed by $0$ is
rejected, so any $0^*11^*0$ is rejected.
Any extension to a rejected string is also rejected (i.e. there is no
route back from a rejecting state to an accepting state), so any
$0^*11^*0(0+1)^*$ is rejected.


This suggests a three state DFA with two accepting states and one rejecting state ...