- WE'RE GIVEN THE GRAPH 
OF A FUNCTION
AND ASKED TO DETERMINE 
THE OPEN INTERVALS
FOR WHICH THE FIRST DERIVATIVE 
IS POSITIVE OR NEGATIVE.
THE FIRST DERIVATIVE IS POSITIVE
IF THE FUNCTION IS INCREASING 
OVER AN INTERVAL
OR IF THE SLOPE OF ANY TANGENT 
LINE OVER THE INTERVAL
HAS A POSITIVE SLOPE.
THE FIRST DERIVATIVE IS NEGATIVE 
OVER AN INTERVAL
IF THE FUNCTION IS DECREASING 
OVER THE INTERVAL
OR IF THE SLOPE OF ANY TANGENT 
LINE OVER THE INTERVAL
WOULD HAVE A NEGATIVE SLOPE.
SO LOOKING AT THIS GRAPH 
FROM LEFT TO RIGHT
NOTICE ON THE LEFT THE FUNCTION 
IS GOING UPHILL
AND THEREFORE IT'S INCREASING 
OVER THIS INTERVAL HERE
UNTIL IT REACHES THIS HIGH POINT 
AT X = -2.
ALSO NOTICE IF WE WERE TO SELECT 
ANY POINT ON THIS INTERVAL
THE SLOPE OF THE TANGENT LINE 
WOULD BE POSITIVE.
THEREFORE THE FIRST DERIVATIVE 
IS POSITIVE
OVER THE OPEN INTERVAL FROM 
-INFINITY TO -2.
BUT THEN NOTICE TO THE RIGHT 
OF -2
THE FUNCTION STARTS TO DECREASE 
OR GO DOWNHILL
FROM LEFT TO RIGHT,
CONTINUES DECREASING UNTIL IT 
REACHES THIS LOW POINT
HERE AT X = +2.
AGAIN IF YOU WERE TO SELECT 
A POINT ON THE GRAPH
OVER THIS INTERVAL 
AND SKETCH A TANGENT LINE,
NOTICE HOW THE SLOPE OF THE 
TANGENT LINE WOULD BE NEGATIVE.
WHICH MEANS THE FIRST DERIVATIVE 
IS NEGATIVE
OVER THE OPEN INTERVAL 
FROM -2 TO +2.
AND THEN TO THE RIGHT OF +2
THE FUNCTION STARTS TO GO 
UP HILL AGAIN
OR INCREASE FROM THE INTERVAL 
FROM 2 TO + INFINITY.
AND ONCE AGAIN IF WE WERE 
TO SKETCH A TANGENT LINE
ANYWHERE OVER THIS INTERVAL
THE SLOPE OF THE TANGENT LINE 
WOULD BE POSITIVE.
SO THE FIRST DERIVATIVE 
IS POSITIVE,
NOT ONLY OVER THIS OPEN INTERVAL
BUT ALSO FROM THE OPEN INTERVAL 
FROM 2 TO INFINITY.
EVEN THOUGH WE DID ANSWER 
THIS QUESTION
I DO WANT TO POINT OUT ONE MORE 
IMPORTANT THING HERE.
NOTICE WHERE 
THE FUNCTION CHANGES
FROM INCREASING TO DECREASING
WE HAVE A HIGH POINT WHICH 
WE CALL A LOCAL MAXIMUM.
THIS OCCURS WHERE THE DERIVATIVE
CHANGES FROM POSITIVE 
TO NEGATIVE
AND THEN WHERE 
THE FUNCTION CHANGES
FROM DECREASING TO INCREASING,
NOTICE HOW WE HAVE A LOW POINT 
OR LOCAL MINIMUM
AND THESE OCCURS WHERE  
THE DERIVATIVE CHANGES
FROM NEGATIVE TO POSITIVE.
I HOPE YOU FOUND THIS HELPFUL.
