After adding another row into the table t1:
INSERT INTO t1 VALUES ('b', 'b');
similar problems with comparison predicates and with BETWEEN easily can be demonstrated:

MariaDB [test]> EXPLAIN
    -> SELECT MIN(f1) FROM t1 WHERE f1 >= 'abc';
+----+-------------+-------+------+---------------+------+---------+------+------+-------------+
| id | select_type | table | type | possible_keys | key  | key_len | ref  | rows | Extra       |
+----+-------------+-------+------+---------------+------+---------+------+------+-------------+
|  1 | SIMPLE      | t1    | ALL  | NULL          | NULL | NULL    | NULL |    2 | Using where |
+----+-------------+-------+------+---------------+------+---------+------+------+-------------+
1 row in set (0.00 sec)

MariaDB [test]> SELECT MIN(f1) FROM t1 WHERE f1 >= 'abc';
+---------+
| MIN(f1) |
+---------+
| b       |
+---------+
1 row in set (0.00 sec)

MariaDB [test]> 
MariaDB [test]> EXPLAIN
    -> SELECT MIN(f2) FROM t1 WHERE f2 >= 'abc';
+----+-------------+-------+------+---------------+------+---------+------+------+------------------------------+
| id | select_type | table | type | possible_keys | key  | key_len | ref  | rows | Extra                        |
+----+-------------+-------+------+---------------+------+---------+------+------+------------------------------+
|  1 | SIMPLE      | NULL  | NULL | NULL          | NULL | NULL    | NULL | NULL | Select tables optimized away |
+----+-------------+-------+------+---------------+------+---------+------+------+------------------------------+
1 row in set (0.00 sec)

MariaDB [test]> SELECT MIN(f2) FROM t1 WHERE f2 >= 'abc';
+---------+
| MIN(f2) |
+---------+
| a       |
+---------+
1 row in set (0.00 sec)

MariaDB [test]> 
MariaDB [test]> EXPLAIN
    -> SELECT MIN(f1) FROM t1 WHERE f1 BETWEEN 'abc' AND 'b' ;
+----+-------------+-------+------+---------------+------+---------+------+------+-------------+
| id | select_type | table | type | possible_keys | key  | key_len | ref  | rows | Extra       |
+----+-------------+-------+------+---------------+------+---------+------+------+-------------+
|  1 | SIMPLE      | t1    | ALL  | NULL          | NULL | NULL    | NULL |    2 | Using where |
+----+-------------+-------+------+---------------+------+---------+------+------+-------------+
1 row in set (0.00 sec)

MariaDB [test]> SELECT MIN(f1) FROM t1 WHERE f1 BETWEEN 'abc' AND 'b' ;
+---------+
| MIN(f1) |
+---------+
| b       |
+---------+
1 row in set (0.00 sec)

MariaDB [test]> 
MariaDB [test]> EXPLAIN
    -> SELECT MIN(f2) FROM t1 WHERE f2 BETWEEN 'abc' AND 'b' ;
+----+-------------+-------+------+---------------+------+---------+------+------+-------------------------+
| id | select_type | table | type | possible_keys | key  | key_len | ref  | rows | Extra                   |
+----+-------------+-------+------+---------------+------+---------+------+------+-------------------------+
|  1 | SIMPLE      | NULL  | NULL | NULL          | NULL | NULL    | NULL | NULL | No matching min/max row |
+----+-------------+-------+------+---------------+------+---------+------+------+-------------------------+
1 row in set (0.00 sec)

MariaDB [test]> SELECT MIN(f2) FROM t1 WHERE f2 BETWEEN 'abc' AND 'b' ;
+---------+
| MIN(f2) |
+---------+
| NULL    |
+---------+
1 row in set (0.00 sec)