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