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CONNECTING F’ AND F’’ WITH
F(X)
SECOND DERIVATIVE
TEST2.1
THE FIRST DERIVATIVE TEST Remember that the first derivative test
will tell where a function is increasing or decreasing.
OBJECTIVES Use the first derivative test to determine
intervals of increase/decrease Use the second derivative test to
determine intervals of concave up/down
CONCAVITY Concavity is the “bend” of the graph
Concave up –-
Concave down--
SECOND DERIVATIVE TEST Points of Inflection- Similar to critical
points. This is where a function changes from concave up to concave down.Find the second derivative and find where it
is equal to zero or is undefinedThese are possible points of inflection
EXAMPLE Perform the first and second derivative
test on and determine the intervals of increase/decrease and concave up/down.
4 3( ) 4f x x x
EXAMPLE Perform the first and second derivative
test on to determine intervals of increase/decrease and concave up/down.
5 3( ) 3 5 3f x x x 5 3( ) 3 5 3f x x x
EXAMPLE Find all points of inflection as well as all
intervals of concave up/down in the function
on the interval [ ]
2( ) 2cos sinf x x x ,
HOMEWORK Pg 228 #31-45 odd find interval of
increase/decrease and concave up and down
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