Concavity  and  Second  Derivative  Test
Definition  of  Concavity
Let  f  be  differentialbel  on  an  open  interval.    The  graph  of  f  is  concave  upward  if  f’  is   increasing  on  the  interval  and  concave  downward  if  f  ’  is  decreasing.
The  definition  references  when  the  ____________________________is  increasing  or   decreasing.
How  can  we  tell  if  the  DERIVATIVE  is  increasing  or  decreasing?
Test  for  Concavity
Let  f  be  a  function  whose  second  derivative  exists  on  an  open  interval
1.
If  f  ‘’(x)>0  then  the  graph  of  f  is  ______________________________________
2.
If  f  ‘’(x)<0  then  the  graph  of  f  is  ____________________________________________
Points  of  Inflection
For  any  value  of  c  that  f  ‘’(c)  =  ______,  or  f  ‘’(c)  is  ______________________  is  a  point  of   inflection.    This  is  where  the  graph  will  change  from  concave  up  to  concave  down.
Second  Derivative  Test
You  can  plug  the  critical  numbers  into  the  second  derivative,  f  ‘’(x).
1.
If  f  ‘’(c)>0  then  that  point  is  a  _________________________________________
2.
If  f  ‘’(c)  <0  then  that  point  is  a  ______________________________________________

Find  the  points  of  inflection  of   f ( x )
= x
4 −
4 x
3
Find  the  relative  extrema  for   f ( x )
= −
3 x 5 +
5 x 3