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X
- IF F OF X IS POSITIVE, DECREASING AND CONCAVE UP
THEN THE ANTIDERIVATIVE FUNCTION BIG F OF X
IS ONE OF THE FOLLOWING:
DECREASING AND CONCAVE UP, DECREASING AND CONCAVE DOWN,
INCREASING AND CONCAVE DOWN OR INCREASING AND CONCAVE UP.
LET'S BEGIN BY REVIEWING THE DEFINITION
OF THE ANTIDERIVATIVE FUNCTION.
BIG F OF X IS AN ANTIDERIVATIVE OF F OF X
IF BIG F PRIME OF X EQUALS F OF X.
SINCE BIG F PRIME OF X EQUALS F OF X
WHEN WE'RE GIVEN INFORMATION ABOUT F OF X HERE
WE CAN THINK OF THIS AS INFORMATION
ABOUT BIG F PRIME OF X.
SO WHEN WE'RE TOLD THAT F OF X IS POSITIVE,
SINCE F OF X EQUALS BIG F PRIME OF X
THAT MEANS BIG F PRIME OF X WOULD HAVE TO BE POSITIVE
AND THEREFORE BIG F OF X WOULD BE INCREASING.
SO AGAIN, F OF X EQUALS BIG F PRIME OF X
WHICH WE KNOW IS POSITIVE OR GREATER THAN ZERO
AND THE SIGN OF THE DERIVATIVE INDICATES WHETHER A FUNCTION
IS INCREASING OR DECREASING
AND SINCE THE DERIVATIVE HERE IS POSITIVE
THIS MEANS BIG F MUST BE INCREASING.
NEXT, WE'RE TOLD THAT F OF X IS DECREASING.
WELL IF F OF X IS DECREASING THEN F PRIME OF X
WOULD HAVE TO BE LESS THAN ZERO OR NEGATIVE.
IF F PRIME OF X IS LESS THAN ZERO
THAT MEANS BIG F DOUBLE PRIME OF X
WOULD ALSO HAVE TO BE LESS THAN ZERO.
SO NOW WE KNOW THAT F PRIME OF X
EQUALS BIG F DOUBLE PRIME OF X WHICH IS LESS THAN ZERO.
THE SECOND DERIVATIVE OF A FUNCTION INDICATES
THE CONCAVITY WHERE IF THE SECOND DERIVATIVE
IS NEGATIVE THE FUNCTION IS CONCAVE DOWN.
IF THE SECOND DERIVATIVE IS POSITIVE
THE FUNCTION IS CONCAVE UP.
BECAUSE THE SECOND DERIVATIVE OF BIG F IS NEGATIVE
OR LESS THAN ZERO THIS MEANS BIG F OF X IS CONCAVE DOWN.
THEN FINALLY, WE'RE TOLD THAT F OF X IS CONCAVE UP.
WELL IF F OF X IS CONCAVE UP
THAT WOULD INDICATE THAT THE SECOND DERIVATIVE OF F
WOULD BE POSITIVE.
SO IF THE SECOND DERIVATIVE OF F IS POSITIVE
THAT MEANS THE THIRD DERIVATIVE OF BIG F
WOULD ALSO BE POSITIVE.
SO F DOUBLE PRIME OF X WHICH EQUALS BIG F
TRIPLE PRIME OF X WOULD BE POSITIVE.
BUT WE DON'T NEED THIS INFORMATION
TO ANSWER THE QUESTION
BECAUSE WE NOW KNOW THAT BIG F IS INCREASING
AND CONCAVE DOWN AND THEREFORE THIS WOULD BE THE ANSWER HERE.
BUT BEFORE WE GO LET'S SKETCH A POSSIBLE GRAPH OF BIG F.
BECAUSE BIG F IS INCREASING WE KNOW AS X INCREASES
THE FUNCTION VALUES INCREASE
AND THEREFORE THE GRAPH WILL BE GOING UPHILL FROM LEFT TO RIGHT
BUT THE FUNCTION IS ALSO CONCAVE DOWN
SO IT MUST RESEMBLE THIS SHAPE HERE.
BUT BECAUSE THE FUNCTION IS ONLY INCREASING
IT WILL ONLY RESEMBLE THE LEFT SIDE
OR THIS SIDE HERE OF THE GRAPH THAT'S CONCAVE DOWN.
WHICH MEANS A POSSIBLE GRAPH FOR BIG F OF X
MIGHT LOOK SOMETHING LIKE THIS.
NOTICE HOW THE FUNCTION IS INCREASING
AND IT'S ALSO CONCAVE DOWN.
I HOPE YOU FOUND THIS HELPFUL.