Prova questa query, il nome del database è test, puoi usare il tuo o rimuoverlo:
SELECT
MIN(rate) AS minRate,
(select date from test.rates where rate = min(co.rate) and
month(date) = month(co.date) and year(date) = year(co.date) limit
)as min_date,
MAX(rate) AS maxRate,
(select date from test.rates where rate = max(co.rate) and
month(date) = month(co.date) and year(date) = year(co.date) limit 1) as
max_date
FROM test.rates co
GROUP BY year(date) , month(date)