Minimal DFA for language of all strings over $\{a,b\}$ whose penultimate symbol is $b$
Note that we have at least three states:


The second last character is $b$ so the tail 2-substring is either $ba$ or $bb$.
The last two characters are $ab$ (read at least one more character).
The last two characters are $aa$ (read at least two more characters).


which leads to the following DFA:


The above DFA contains minimal number of states if and only if it equals its minimization. Following this page in DFA Minimization using Equivalence Theorem we have

$P_0 = \{(1, 2), (3)\}\\
P_1 = \{(1), (2), (3)\}$

In $P_1$ we used the fact that one single character may lead state 2 to an accepting state, whereas 1 does not "reach that far". Since $P_1$ gives no opportunity for further refinement, we may stop and we know that the DFA is minimized.