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- NOW WE'LL LOOK AT SEVERAL EXAMPLES OF DETERMINING
WHETHER THE FUNCTION VALUE, FIRST DERIVATIVE FUNCTION VALUE,
AND SECOND DERIVATIVE FUNCTION VALUE
ARE POSITIVE, NEGATIVE OR ZERO AT A GIVEN POINT.
LOOKING AT OUR FIRST EXAMPLE,
NOTICE HOW THE POINT IS ABOVE THE X AXIS
AND THEREFORE THE Y VALUE OR FUNCTION VALUE MUST BE POSITIVE.
SO F OF X IS POSITIVE OR GREATER THAN ZERO.
NEXT, THE FIRST DERIVATIVE
GIVES US THE SLOPE OF THE TANGENT LINE.
NOTICE IS WE WERE TO SKETCH A TANGENT LINE AT THIS POINT
THE SLOPE OF THE TANGENT LINE WOULD BE POSITIVE
AND THEREFORE THE FIRST DERIVATIVE IS POSITIVE.
SO F PRIME OF X IS GREATER THAN ZERO OR AGAIN POSITIVE.
ALSO NOTICE THAT THE GIVEN FUNCTION IS INCREASING
ON THIS INTERVAL
THAT CONTAINS THE GIVEN POINT
AND THEREFORE THE FIRST DERIVATIVE WOULD BE POSITIVE.
NOW FOR THE SECOND DERIVATIVE WHICH INDICATES CONCAVITY
NOTICE HOW THE FUNCTION IS CONCAVE DOWN ON THIS INTERVAL
THAT CONTAINS THE GIVEN POINT
AND THEREFORE THE SECOND DERIVATIVE
WOULD HAVE TO BE NEGATIVE AT THIS POINT.
SO F DOUBLE PRIME OF X IS LESS THAN ZERO
OR WE CAN SAY IT'S NEGATIVE.
LOOKING AT THE NEXT GRAPH,
ONCE AGAIN NOTICE HOW THE POINT IS ABOVE THE X AXIS
AND THEREFORE THE Y VALUE OR FUNCTION VALUE IS POSITIVE.
SO F OF X IS GREATER THAN ZERO.
NEXT, FOR THE FIRST DERIVATIVE
IF WE SKETCH A TANGENT LINE AT THIS POINT
NOTICE HOW THE SLOPE OF THE TANGENT LINE WOULD BE NEGATIVE
AND THEREFORE THE FIRST DERIVATIVE WOULD BE NEGATIVE.
SO F PRIME OF X IS LESS THAN ZERO.
ALSO NOTICE THAT THE FUNCTION IS DECREASING
ON THIS INTERVAL HERE THAT CONTAINS THE POINT
AND THEREFORE THE FIRST DERIVATIVE WOULD BE NEGATIVE,
AND NOW FOR THE SECOND DERIVATIVE
NOTICE HOW THE GRAPH IS CONCAVE DOWN ON THIS INTERVAL
WHICH CONTAINS THE GIVEN POINT
AND THEREFORE THE SECOND DERIVATIVE IS NEGATIVE
OR LESS THAN ZERO.
FOR OUR NEXT EXAMPLE NOTICE HOW THE POINT IS BELOW THE X AXIS
AND THEREFORE THE FUNCTION VALUE OR Y VALUE WOULD BE NEGATIVE.
SO F OF X IS LESS THAN ZERO OR NEGATIVE.
FOR THE FIRST DERIVATIVE,
NOTICE HOW IF WE SKETCH A TANGENT LINE AT THIS POINT
THE SLOPE OF THE TANGENT LINE IS NEGATIVE
THEREFORE THE FIRST DERIVATIVE IS NEGATIVE OR LESS THAN ZERO.
ALSO NOTICE ON THIS INTERVAL HERE THE FUNCTION IS DECREASING
AND THE POINT IS IN THIS INTERVAL
AND THEREFORE THE FIRST DERIVATIVE WOULD BE NEGATIVE
AND NOW FOR THE SECOND DERIVATIVE,
NOTICE HOW THE FUNCTION IS CONCAVE UP ON THIS INTERVAL
THAT CONTAINS THE GIVEN POINT
AND THEREFORE THE SECOND DERIVATIVE WOULD BE POSITIVE.
SO F DOUBLE PRIME OF X IS GREATER THAN ZERO, POSITIVE.
OUR NEXT EXAMPLE AGAIN,
NOTICE HOW THE POINT IS BELOW THE X AXIS
AND THEREFORE THE Y VALUE OR FUNCTION VALUE IS NEGATIVE.
SO F OF X IS LESS THAN ZERO.
FOR THE FIRST DERIVATIVE,
IF WE SKETCH A TANGENT LINE AT THIS POINT
NOTICE HOW THE SLOPE OF THE TANGENT WOULD BE POSITIVE
AND THEREFORE THE FIRST DERIVATIVE IS POSITIVE
OR GREATER THAN ZERO,
AND THEN FOR THE SECOND DERIVATIVE,
THE FUNCTION IS CONCAVE UP ON THIS INTERVAL
THAT CONTAINS THE GIVEN POINT
AND THEREFORE THE SECOND DERIVATIVE WOULD BE POSITIVE.
NOW FOR OUR LAST TWO EXAMPLES,
NOTICE HOW THE GIVEN POINT IS ON THE X AXIS.
EVERY POINT ON THE X AXIS HAS A Y COORDINATE
OR FUNCTION VALUE OF ZERO
AND THEREFORE F OF X IS EQUAL TO ZERO.
FOR F PRIME OF X
NOTICE HOW THE SLOPE OF THE TANGENT LINE AT THIS POINT
WOULD HAVE A NEGATIVE SLOPE
AND THEREFORE THE FIRST DERIVATIVE IS NEGATIVE.
IT'S ALSO TRUE THAT THE FUNCTION IS DECREASING ON THIS INTERVAL
CONTAINING THE GIVEN POINT
AND THEREFORE THE FIRST DERIVATIVE IS NEGATIVE
AND THEN FOR THE SECOND DERIVATIVE,
NOTICE HOW THE FUNCTION IS CONCAVE DOWN ON THIS INTERVAL
AND CONCAVE UP ON THIS INTERVAL.
SO THE FUNCTION CHANGES CONCAVITY AT THIS GIVEN POINT
WHICH MEANS THE POINT IS A POINT OF INFLECTION
AND THEREFORE THE SECOND DERIVATIVE
WOULD HAVE TO BE ZERO,
AND FOR OUR LAST EXAMPLE,
NOTICE HOW THE GIVEN POINT IS BELOW THE X AXIS
AND THEREFORE THE Y VALUE OR FUNCTION VALUE IS NEGATIVE.
FOR THE FIRST DERIVATIVE
IF WE SKETCH A TANGENT LINE AT THIS POINT
NOTICE HOW THE SLOPE OF THE TANGENT LINE WOULD BE ZERO
BECAUSE WE HAVE A HORIZONTAL TANGENT LINE.
SO THE FIRST DERIVATIVE WOULD BE ZERO
REMEMBER THIS ALSO TELLS US
THAT THIS X VALUE WOULD BE A CRITICAL NUMBER,
AND THEN FINALLY FOR THE SECOND DERIVATIVE,
NOTICE HOW THE FUNCTION IS CONCAVE UP ON THIS INTERVAL
AND THEREFORE THE SECOND DERIVATIVE WOULD BE POSITIVE.
I HOPE YOU FOUND THIS HELPFUL.