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- 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.