sqlite3 select min, max together is much slower than select them separately

Question

sqlite> explain query plan select max(utc_time) from RequestLog;
0|0|0|SEARCH TABLE RequestLog USING COVERING INDEX key (~1 rows) # very fast

sqlite> explain query plan select min(utc_time) from RequestLog;
0|0|0|SEARCH TABLE RequestLog USING COVERING INDEX key (~1 rows) # very fast

sqlite> explain query plan select min(utc_time), max(utc_time) from RequestLog;
0|0|0|SCAN TABLE RequestLog (~8768261 rows) # will be very very slow

While I use min and max separately, it works perfectly. However, sqlite will 'forget' the index while I select the min and max together for some reason. Is there any configuration I can do (I used Analyze already, it won't work)? or is there any explanation for this behavior?

EDIT1

sqlite> .schema 

CREATE TABLE FixLog(
                    app_id text,  __key__id INTEGER,
                    secret text, trace_code text, url text,
                    action text,facebook_id text,ip text,
                    tw_time datetime,time datetime,
                    tag text,to_url text,
                    from_url text,referer text,weight integer,
                    Unique(app_id, __key__id)
                    );
CREATE INDEX key4 on FixLog(action);
CREATE INDEX time on FixLog(time desc);
CREATE INDEX tw_time on FixLog(tw_time desc);



sqlite> explain query select min(time) from FixLog;
0|0|0|SEARCH TABLE FixLog USING COVERING INDEX time (~1 rows)
sqlite> explain query select max(time) from FixLog;
0|0|0|SEARCH TABLE FixLog USING COVERING INDEX time (~1 rows)
sqlite> explain query plan select max(time), min(time) from FixLog;
0|0|0|SCAN TABLE FixLog (~1000000 rows)

Answer 1

This is a known quirk of the sqlite query optimizer, as explained here: http://www.sqlite.org/optoverview.html#minmax:

Queries of the following forms will be optimized to run in logarithmic time assuming appropriate indices exist:

 SELECT MIN(x) FROM table;
 SELECT MAX(x) FROM table;

In order for these optimizations to occur, they must appear in exactly the form shown above - changing only the name of the table and column. It is not permissible to add a WHERE clause or do any arithmetic on the result. The result set must contain a single column. The column in the MIN or MAX function must be an indexed column.

UPDATE (2017/06/23): Recently, this has been updated to say that a query containing a single MAX or MIN might be satisfied by an index lookup (allowing for things like arithmetic); however, they still preclude having more than one such aggregation operator in a single query (so MIN,MAX will still be slow):

Queries that contain a single MIN() or MAX() aggregate function whose argument is the left-most column of an index might be satisfied by doing a single index lookup rather than by scanning the entire table. Examples:

SELECT MIN(x) FROM table;
SELECT MAX(x)+1 FROM table;