CURVE SKETCHING
(Critical Points)
Points on a function where f ‘(x) = 0 or where f ‘(x) does not exist
help to define the shape of the function’s graph so they are called
critical points. A critical number is any value, c, in the domain of
f(x), at which f ‘(c)= 0 or where f ‘(c) does not exist (DNE).
The First Derivative Test for Critical Points of a Function
Let c be the critical number of a continuous function.
c
c
f '(x)0
f '(x)0
If f ‘(x) changes from positive (increasing)
to negative (decreasing) at c, then f(x) has
a local maximum at c.
f '(x)0
If f ‘(x) changes from negative (decreasing)
to positive (increasing) at c, then f(x) has
a local minimum at c.
c
f '(x)0
f '(x)0
c
f '(x)0
f '(x)0
f '(x)0
If f ‘(x) does not change sign at c, then f(x) has neither a local maximum nor a local
minimum at that point. This is called a point of inflection (POI).
Algorithm for Analyzing Critical Points:
1.
2.
3.
Determine the critical numbers of the function (where f ‘(x) = 0 or f ‘(x) DNE ).
Use the first derivative to determine whether f(x) is increasing or decreasing
on either side of each critical number.
Conclude whether each critical point is a local extrema, POI, or neither.
Example 
a)
Determine all critical points and sketch the function.
f( x ) = x3 – 3x2 – 9x + 27
y
x
f ‘(x)
b)
f( x ) = x4 – 4x3
y
x
f ‘(x)
c)
f ( x )  ( x  2)
2
3
y
x
f ‘(x)
Homework:
p.178–180
#2, 3, 5, 7–15