SQL Selección de fechas con máxima venta para cada departamento

Puedes probar a continuación-

with cte as 
(
 SELECT 
 Departments.Name, SALES.Date_sale, SUM(GOODS.Price * SALES.Quantity) 
 AS profit FROM DEPARTMENTS inner join GOODS on DEPARTMENTS.Dept_id = GOODS.Dept_id
 inner join SALES on GOODS.Good_id = SALES.Good_id
 GROUP BY DEPARTMENTs.Name, SALES.Date_sale
)A

select * from cte a
where profit =
     (select max(profit) from cte b on a.department=b.department)

O puede usar row_number()

select * from
(
select *, row_number() over(partition by department oder by profit desc) as rn
from cte
)A where rn=1