- IF F PRIME OF X IS NEGATIVE 
DECREASING IN CONCAVE UP,
THEN THE FUNCTION F OF X IS ONE 
OF THE FOLLOWING,
DECREASING IN CONCAVE UP,
DECREASING IN CONCAVE DOWN,
INCREASING IN CONCAVE DOWN,
OR INCREASING IN CONCAVE UP.
THE SIGN OF THE FIRST DERIVATIVE 
INDICATES WHETHER A FUNCTION
IS INCREASING OR DECREASING.
IF THE FIRST DERIVATIVE 
IS POSITIVE,
THE FUNCTION IS INCREASING.
IF THE FIRST DERIVATIVE 
IS NEGATIVE,
THE FUNCTION IS DECREASING.
AS WE'RE TOLD THAT F PRIME OF X 
IS NEGATIVE, OR LESS THAN ZERO,
THIS TELLS US THAT THE FUNCTION 
F OF X IS DECREASING.
NEXT, F PRIME OF X IS DECREASING
WHICH MEANS A DERIVATIVE 
OF F PRIME OF X,
OR THE SECOND DERIVATIVE 
WOULD BE NEGATIVE.
AND THE SECOND DERIVATIVE
INDICATES A CONCAVITY 
OF THE FUNCTION
WHERE IF THE SECOND DERIVATIVE 
IS POSITIVE,
THE FUNCTION IS CONCAVE UP.
BUT IN OUR CASE BECAUSE THE 
SECOND DERIVATIVE IS NEGATIVE,
THIS INDICATES THE FUNCTION F 
OF X IS CONCAVE DOWN.
WE'RE ALSO TOLD F PRIME OF X 
IS CONCAVE UP,
WHICH MEANS THE SECOND 
DERIVATIVE OF THE DERIVATIVE
OR THE THIRD DERIVATIVE 
WOULD BE POSITIVE.
BUT THIS ACTUALLY 
DOES NOT ANSWER THE QUESTION.
WE NOW KNOW THAT F OF X 
IS DECREASING AND CONCAVE DOWN
AND THEREFORE OUR ANSWER 
IS THE SECOND CHOICE HERE.
NOW, LETS SEE IF WE CAN SKETCH 
A POSSIBLE GRAPH FOR F OF X.
WELL, IF F OF X IS DECREASING
WE KNOW FROM LEFT TO RIGHT
THE GRAPH IS GOING DOWNHILL 
OR AS X INCREASES Y DECREASES,
AND IF IT'S CONCAVE DOWN,
IT MUST RESEMBLE AT LEAST
A PIECE OF A GRAPH
THAT LOOKS LIKE THIS.
AND BECAUSE F OF 
X IS ALWAYS DECREASING
IT WOULD HAVE TO RESEMBLE 
THE RIGHT SIDE OF THIS PIECE
THAT IS CONCAVE DOWN.
SO THE GRAPH MIGHT LOOK 
SOMETHING LIKE THIS.
NOTICE HOW IT IS DECREASING 
AND ALSO CONCAVE DOWN.
I HOPE YOU HAVE FOUND THIS 
HELPFUL.
