Valores mínimos y máximos de Mysql y la fecha correspondiente para cada mes

Pruebe esta consulta, el nombre de la base de datos es prueba, puede usar el suyo o eliminarlo:

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)