Chapter 4 Section 2
 Prerequisites for Calculus

Second Derivative Test for Concavity

y=f(x) is concave up where y '' > 0
y=f(x) is concave down where y '' < 0

Point of Inflection

Change in concavity
(where the 2nd derivative changes signs)

Point of inflection is found when f ''(x) = 0 or undefined.
 

Ex: y = x3 - 6x2 + 9x + 1
Find critical points:
    y '  = 3x2 - 12x + 9
     (x-1)(x-3)
    (1,5) and (3,1)

Point of inflection and concavity:
y '' = 6x - 12
     x = 2  then (2,3) may be a point of inflection if concavity changes.

Check concavity:
y ''(1) = -6 concave down
y ''(3) = 6 concave up
therefore: (2,3) is a
point of inflection.