4.4 Concavity and Inflection
Points
Wed Dec 3
Do Now
Find the 2nd derivative of each function
1) f (x) = 3x 2 + 2x - 4
2) f (x) = x 4 + 3x 3 - 5x + 2
Applications of the 2nd
derivative
• So far, we’ve only talked about one
application of the 2nd derivative, which
is the acceleration function
• The second derivative can also be used
to describe the behavior of functions as
well.
Concavity and Inflections
• The 1st derivative is used to describe
slope. But since it is also a function, it
also has its own “slope” or derivative.
• The 2nd derivative can be used to
model the behavior of the slope, as it is
ALSO changing with the function
– Some slopes can be steep, while others
rather flat
Concavity
• The 2nd derivative can be used to
describe concavity
• Concavity is the rate at which the slope
increases or decreases
• There are two types of concavity
– Concave up (looks like a smile)
– Concave down (looks like a frown)
Concavity and f’’(x)
• Thm- Suppose f(x) is differentiable on
an interval I and f ’’(x) exists,
– If f ’’(x) > 0, then the graph is concave up
– If f ’’(x) < 0, then the graph is concave
down
Note: Second derivative only
Inflection Points
• An inflection point is a point on the
graph where a graph alternates
between concave up and concave down
• We can find inflection points when
f ’’(x) = 0
Example 1
• Determine where the graph is concave
up and concave down
f (x) = 2x + 9x - 24 x -10
3
2
Example 2
• Determine where the graph is concave
up and down, and find any inflection
4
2
points
f (x) = x - 6x +1
Ex 5.3
• Determine the concavity and inflection
4
points of
f (x) = x
2nd Derivative Test
• The 2nd Derivative can also be used to
determine if a critical point is a local max or
min.
• Thm- Suppose that f(x) is continuous on
an interval (a,b) and f’( c) = 0, then
– If f’’( c) < 0, then c is a local max
• Concave down means a local max!
– If f’’( c) > 0, then c is a local min
• Concave up means a local min!
Warning!
• The 2nd derivative test does not always work.
• It will not work if f’’(c) = 0
• If the 2nd derivative test does not work, you
must use the table
Closure
• Journal Entry: Find the points of
inflection and intervals of concavity of
f (x) = 3x 5 - 5x 4 +1
• HW: p.238 #1-55 odds (every other
odd) 4.2-4.4 quiz monday
4.4 2nd Derivative Test
Thurs Dec 4
• Do Now
• Find the intervals of concavity and
inflection points of the following function
f (x) = x + 4x -1
4
3
HW Review p.238 first half
Ex
• Analyze the critical points of
f (x) = x - 8x +1
4
2
Ex 2
• Use the 2nd derivative test to find the local
max and mins for
5
3
f (x) = x - x
Worksheet
Closure:
• Hand in: Find intervals of increase,
decrease, and concavity, local extrema,
and inflection points of
f (x) = x + 4 x +1
4
2
• HW: p.239 #1-55 odds all the other
odds 4.2-4.4 Quiz Monday
4.2-4.4 Review
Fri Dec 5
• Do Now
• Find all intervals of increase/decrease, local
extrema, intervals of concavity, and inflection
points for the function
2 -x
f (x) = x e
4.2-4.4 Review for Quiz
• 1st Derivative Info
– Max and Min (critical points)
– Absolute max/mins
– Increasing / Decreasing Intervals
• 2nd Derivative Info
– Inflection Points
– Concavity Intervals
• Don’t forget your derivative rules!
– Product, quotient, chain, etc
– Interpret/Draw Graphs - no graphing calc
The 2 derivatives and the
relationship between graphs
Closure
• Journal Entry: What is concavity? What
does the 2nd derivative tell us about the
original function?
• 4.2-4.4 Quiz Mon