# MAT135H1 Lecture Notes - Lecture 3: Inflection, If And Only If

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4.4 Concavity and Points of Inflection

• A function

()fx

is concave up on an interval if

''( ) 0fx>

o The graph of the function is above the tangent on every point of the interval

• A function

()fx

is concave down on an interval if

''( ) 0fx<

o The graph of the function is below the tangent on every point of the interval

• Point of Inflection:

o a point on a graph where

()fx

changes from concave up to concave down and vice versa

o a point of inflection occurs when

''( ) 0fx=

or undefined

• Second Derivative Test:

o another test used to find local max or min values

! find critical points using

'( ) 0fx=

! find

''( )fx

if it exists

! evaluate

''( )fx

at critical points

• if

''( ) 0fc>

, then the critical point is a local min

• if

''( ) 0fc<

, then the critical point is a local max

• if

''( ) 0fc=

, then the test fails; use the first derivative test