[ MUSIC ]
- WELCOME TO THE FIRST 
DERIVATIVE TEST.
THE GOAL OF THIS VIDEO 
WILL BE TO FIND
RELATIVE EXTREMA OF A FUNCTION 
USING THE FIRST DERIVATIVE.
THIS PRESENTATION ASSUMES 
YOU'VE ALREADY WATCHED THE VIDEO
ENTITLED "INCREASING 
AND DECREASING FUNCTIONS."
LET'S GO AHEAD AND GET STARTED.
THE FIRST DERIVATIVE TEST 
FOR RELATIVE EXTREMA,
F OF X HAS A RELATIVE MINIMUM 
AT C
IF F OF X IS DECREASING 
TO THE LEFT OF C
AND INCREASING TO THE RIGHT.
SO IF THE FUNCTION IS GOING 
DOWN-HILL AND THEN UP-HILL
TO THE LEFT AND RIGHT OF C,
WE'LL CALL THIS C,
YOU WOULD HAVE TO HAVE A LOW 
POINT OR A RELATIVE MINIMUM.
FROM THE PREVIOUS VIDEO WE ALSO 
KNOW THAT THIS WOULD OCCUR
WHERE THE DERIVATIVE CHANGES 
FROM NEGATIVE TO POSITIVE.
NEGATIVE DERIVATIVE MEANS 
THE FUNCTION IS DECREASING.
POSITIVE DERIVATIVE MEANS 
IT'S INCREASING.
AND IF THE OPPOSITE OCCURS,
IF A FUNCTION CHANGES FROM 
INCREASING TO DECREASING AT C
WE'D HAVE A HIGH POINT 
OR RELATIVE MAXIMUM.
AND, AGAIN, THE DERIVATIVE WOULD 
CHANGE FROM POSITIVE TO NEGATIVE
IN THIS CASE.
AND SOMETIMES THE SIGN OF THE 
FIRST DERIVATIVE WON'T CHANGE.
IF YOU TAKE A LOOK 
AT THIS FUNCTION,
IT'S INCREASING 
AND THEN INCREASING AGAIN.
SO WE'D HAVE A POSITIVE 
DERIVATIVE TO THE LEFT
AND RIGHT OF C.
AND IN THIS CASE WE HAVE NEITHER 
RELATIVE MAX OR A MIN.
SO THIS IS 
WHAT WE'RE GOING TO DO
TO FIND THE RELATIVE EXTREMA 
USING THE FIRST DERIVATIVE.
STEP ONE, WE ARE GOING TO LOCATE 
ANY CRITICAL VALUES
OF THE FUNCTION.
AND WE USE THESE NUMBERS TO 
DETERMINE OUR TEST INTERVALS.
NUMBER TWO, WE'LL DETERMINE THE 
SIGN OF THE FIRST DERIVATIVE.
AND AS WE LEARNED 
FROM A PREVIOUS VIDEO,
IF THE FIRST DERIVATIVE 
IS POSITIVE,
IT'S INCREASING ON THE INTERVAL.
AND IF IT'S NEGATIVE 
IT'S DECREASING.
STEP FOUR, BASED UPON THE SIGN 
CHANGES OF THE DERIVATIVE
WILL DETERMINE 
IF ANY RELATIVE EXTREMA EXIST.
AND THEN IF THE RELATIVE EXTREMA 
EXIST
WE'LL EVALUATE THE ORIGINAL 
FUNCTION AT THE GIVEN X VALUE
TO FIND THE RELATIVE MAX OR MIN.
LET'S GO AHEAD 
AND GIVE IT A TRY.
DETERMINE ANY RELATIVE EXTREMA 
USING THE FIRST DERIVATIVE.
SO STEP ONE WE'RE GOING TO FIND 
THE DERIVATIVE
TO FIND THE CRITICAL NUMBERS.
THE DERIVATIVE WOULD BE EQUAL 
TO 3/4X SQUARED - 3.
THERE ARE NO VALUES OF X 
WHERE THIS DID NOT EXIST,
SO WE'LL SET IT EQUAL TO ZERO 
AND SOLVE.
TO CLEAR THE FRACTIONS HERE,
WHAT WE COULD DO 
IS MULTIPLY 3 BY 4.
IF WE DID THAT WE WOULD OBTAIN 
3X SQUARED - 12 = 0.
FACTOR OUT THE COMMON FACTOR 
OF 3,
FACTOR THE DIFFERENCE 
OF SQUARES.
LOOKS LIKE WE HAVE 
TWO CRITICAL NUMBERS.
WE HAVE X = 2 AND X = -2.
SO THE DOMAIN OF THE ORIGINAL 
FUNCTION IS ALL REALS.
WE'RE GOING TO HAVE 
THREE INTERVALS TO TEST.
HERE THEY ARE FROM -INFINITY 
TO -2, FROM -2 TO 2,
AND FROM 2 TO INFINITY.
LET'S PICK OUR TEST VALUES.
TO DETERMINE THE SIGN 
OF THE FIRST DERIVATIVE
I'M GOING TO GO AHEAD AND USE 
THE GRAPHING CALCULATOR AGAIN.
WE WILL TYPE IN THE DERIVATIVE 
FUNCTION INTO Y1.
REMEMBER THE DERIVATIVE 
WAS 3/4X SQUARED - 3.
USING THE TABLE FEATURE
WE CAN FIND THE SIGN 
OF THE FIRST DERIVATIVE.
IF WE HIT SECOND GRAPH WE CAN 
TYPE IN OUR TEST VALUES OF -3,
0, AND 3.
THE FIRST INTERVAL'S POSITIVE, 
THE SECOND IS NEGATIVE,
AND THE THIRD IS POSITIVE.
LET'S GO AHEAD 
AND RECORD THAT INFORMATION.
SO WHAT WE KNOW IS THAT IT'S 
INCREASING ON THIS INTERVAL,
DECREASING, AND INCREASING.
NOW, THAT'S NOT WHAT THEY ASK US 
HERE.
BUT IF IT CHANGES 
FROM INCREASING TO DECREASING
WE'D HAVE A RELATIVE MAX 
AT X = -2.
AND IF IT CHANGES FROM 
DECREASING TO INCREASING HERE
WE'D HAVE A LOW POINT 
OR RELATIVE MINIMUM AT +2.
SO LET'S GO AHEAD AND RECORD 
THIS DOWN BELOW.
AGAIN, A RELATIVE MAX IS AT 
X = -2
AND OUR MINIMUM WAS AT X = 2.
NOW, THE ACTUAL RELATIVE MAX 
AND MIN ARE THE Y VALUES
OF THE FUNCTION
AT THESE VALUES OF X.
SO WE ACTUALLY HAVE TO FIND 
F OF 2 AND F OF -2.
SO WE HAVE TO EVALUATE 
THE ORIGINAL FUNCTION
AT THESE VALUES OF X.
OUR ORIGINAL FUNCTION WAS 
X CUBED DIVIDED BY 4 - 3X.
SO NOW USING THE TABLE FEATURE 
AGAIN
WE'LL EVALUATE THIS AT -2 AND 2.
SO AT -2 THE Y VALUE IS 4, 
A +2 WHICH IS EQUAL TO -4.
SO OUR RELATIVE MAX = 4 
AT X = -2.
OUR RELATIVE MINIMUM = -4 
AT X = +2.
SO LET'S VERIFY THIS 
WITH A GRAPH.
AND YOU CAN SEE 
HERE'S OUR RELATIVE MAX,
HERE'S OUR RELATIVE MIN. 
LOOKS LIKE OUR WORK IS GOOD.
I HOPE THAT HELPS EXPLAIN 
HOW TO FIND THE RELATIVE EXTREMA
USING THE FIRST DERIVATIVE.
THANK YOU FOR WATCHING.
