- WE WANT TO FIND THE INTERVALS
FOR WHICH THE FUNCTION IS INCREASING AND DECREASING,
AND ALSO DETERMINE ANY RELATIVE EXTREMA.
FORMERLY, A FUNCTION IS INCREASING
IF AS X INCREASES Y INCREASES.
AND A FUNCTION IS DECREASING, IF AS X INCREASES Y DECREASES.
BUT I THINK AN EASIER WAY TO DETERMINE
WHERE A FUNCTION IS INCREASING OR DECREASING
IS TO TRACE THE GRAPH FROM LEFT TO RIGHT,
AND AS WE TRACE THE GRAPH WE'RE MOVING UP-HILL,
THE FUNCTION IS INCREASING.
IF WE'RE MOVING DOWN-HILL, THE FUNCTION IS DECREASING.
SO IF WE START ON THE LEFT OR WE START HERE,
AS WE TRACE THE GRAPH FROM LEFT TO RIGHT
NOTICE HOW WE'RE GOING UP-HILL,
AND THEREFORE THE GRAPH IS INCREASING,
CONTINUES TO INCREASE UNTIL WE REACH THIS HIGHPOINT HERE.
LET'S GO AHEAD AND STOP HERE FOR A MOMENT.
THIS HIGHPOINT REPRESENTS A RELATIVE MAXIMUM.
LET'S GO AHEAD AND FIND THE COORDINATES OF THIS POINT.
THE COORDINATES WOULD BE (-2,18).
NOTICE THE PIECE WE JUST HIGHLIGHT,
AS X INCREASES THE FUNCTION DIE IS ALSO INCREASE.
AND ONCE YOU REACH THIS HIGHPOINT
AND START TO COME DOWN THE OTHER SIDE,
WE'RE MOVING DOWN-HILL,
AND THEREFORE THE FUNCTION IS DECREASING ON THIS INTERVAL.
TO USE A DECREASE UNTIL WE REACH THIS LOW POINT HERE,
WHICH WOULD BE A RELATIVE MINIMUM.
LET'S GO AHEAD AND FIND THE COORDINATES HERE.
COORDINATES WOULD BE (+2,-14).
AS WE MOVE PAST THIS LOW POINT,
NOTICE HOW WE START TO MOVE UP-HILL AGAIN,
AND THEREFORE THE FUNCTION IS INCREASING ON THIS INTERVAL.
NICOTINE USED TO INCREASE.
WE TAKE A LOOK AT THIS MIDDLE REGION JUST FOR A MOMENT,
NOTICE AS X INCREASES Y DECREASES,
AND THAT'S MORE FORMALLY
WHY WE SAY THE FUNCTION IS DECREASING ON THIS INTERVAL.
THEN, AGAIN, AS X INCREASES Y INCREASES
ON THIS LAST INTERVAL,
AND THEREFORE IT'S INCREASING.
NOTICE HOW THE FUNCTION CHANGES
FROM INCREASING TO DECREASING AT X = -2,
AND THEN FROM DECREASING TO INCREASING AT X = +2.
THE LAST THING TO REMEMBER IS,
AS YOU MOVE RIGHT ON THE X AXIS
WE APPROACH POSITIVE INFINITY,
AS WE MOVE LEFT WE APPROACH NEGATIVE INFINITY.
NOW WE HAVE ALL THE INFORMATION WE NEED
TO DETERMINE WHERE THE FUNCTION IS INCREASING
OR DECREASING,
AND LIST ANY RELATIVE EXTREMA.
THE FUNCTION IS INCREASING FROM NEGATIVE INFINITY TO -2,
SO WE USE INTERVALS ON THE X AXIS
TO DETERMINE WHERE THE FUNCTION IS INCREASING
OR DECREASING.
AND WE'RE NOT GOING TO INCLUDE -2 ON THE INTERVAL
BECAUSE AT -2 THE FUNCTION CHANGES
FROM INCREASING TO DECREASING.
AND THAT'S WHERE THE RELATIVE MAXIMUM OCCURS.
WELL, THE FUNCTION IS ALSO INCREASING ON THE INTERVAL
FROM 2 TO INFINITY,
AGAIN, NOT INCLUDING+2.
WHEN THE FUNCTION IS DECREASING
ON THE OPEN INTERVAL FROM -2 TO +2.
THIS HIGHPOINT REPRESENTS A RELATIVE MAXIMUM,
WHICH WE CAN ALSO CALL LOCAL MAXIMUM.
AND THIS LOW POINT REPRESENTS A RELATIVE MINIMUM,
WHICH WE CAN ALSO CALL A LOCAL MINIMUM.
SO LOCAL MAXIMUM IS EQUAL TO THE FUNCTION VALUE OR Y VALUE.
SO LOCAL MAXIMUM OR RELATIVE MAXIMUM
IS = TO 18 WHEN X = -2.
AND THE LOCAL MINIMUM OR RELATIVE MINIMUM IS = TO -14,
THE FUNCTION VALUE AT THE LOW POINT AT X = +2.
THE LAST THING I DO WANT TO MENTION IS,
YOU MAY BE ASKED TO EXPRESS YOUR INTERVALS
USING INEQUALITIES INSTEAD OF INTERVAL NOTATION.
SO LET'S GO AHEAD AND SHOW THAT JUST IN CASE.
THE INTERVAL FROM NEGATIVE INFINITY TO -2
WOULD BE X IS LESS THAN -2,
OR THE INTERVAL FROM 2 TO INFINITY
WOULD BE X IS GREATER THAN 2.
THE OPEN INTERVAL FROM -2 TO 2 CAN BE WRITTEN
AS X IS GREATER THAN -2 AND LESS THAN 2.
AND I THINK WE'LL GO AHEAD AND STOP THERE.
I HOPE YOU FOUND THIS HELPFUL.