This looks like T-SQL so I'll give the answer for that.
SELECT a.id,
a.first_name,
a.last_name,
MIN(b.income),
(SELECT TOP 1 c.departmentname --Or whatever the name of your department name is
FROM department c
WHERE c.income = MIN(b.income)) AS [DepartmentName]
FROM CUSTOMERS a
INNER JOIN department b ON a.id = b.id
GROUP BY a.id, a.first_name, a.last_name;
You need to use a nested query in order to find which department has the income. You might also have to add in some more where
restraints on the nested query there, assuming multiple departments can have the same income. But those will depend on your database schema, so I'll leave you to work out that logic to make sure you're talking about the same one.
Edit:
Although reading this more, it looks like you could just rephrase it all:
SELECT a.id,
a.first_name,
a.last_name,
(SELECT TOP 1 departmentname --Or whatever the name of your department name is
FROM department
WHERE department.id = customers.id
ORDER BY income DESC) AS [DepartmentName]
FROM customers
You wouldn't get the income with that, but you can add in the code to get that too.