1. Concavity and Second Derivative Test
Lesson 4.4

2. Concavity Concave UP Concave DOWN
Inflection point: Where concavity changes 

3. Inflection Point
Consider the slope as curve changes through concave up to concave down
At inflection point slope reaches maximum positive value
After inflection point, slope becomes less positive 
Slope starts negative
Graph of the slope
Slope becomes positive, then more positive
Becomes less negative
Slope becomes (horizontal) zero

4. Inflection Point
What could you say about the slope function when the original function has an inflection point
Slope function has a maximum (or minimum
Thus second derivative = 0
Graph of the slope

5. Second Derivative This is really the rate of change of the slope
When the original function has a relative minimum
Slope is increasing (left to right) and goes through zero
Second derivative is positive
Original function is concave up

6. Second Derivative When the original function has a relative maximum
The slope is decreasing (left to right) and goes through zero
The second derivative is negative
The original function is concave down
View Geogebra
Demo

7. Not an inflection point
Second Derivative
If the second derivative f ’’(x) = 0
The slope is neither increasing nor decreasing
If f ’’(x) = 0 at the same place f ’(x) = 0
The 2nd derivative test fails
You cannot tell what the function is doing
Not an inflection point

8. Example
Consider
Determine f ‘(x) and f ’’(x) and when they are zero

9. Example
f ’(x) = 0, f’’(x) > 0, this is concave up, a relative minimum
f ‘(x)
f ‘(x) = 0, f ‘’(x) < 0 this is concave down, a maximum
f(x) 
f ‘’(x) = 0 this is an inflection point
f ‘’(x)

10. Example Try f ’(x) = ? f ’’(x) = ?
Where are relative max, min, inflection point?

11. Algorithm for Curve Sketching
Determine critical points
Places where f ‘(x) = 0
Plot these points on f(x)
Use second derivative f’’(x) = 0
Determine concavity, inflection points
Use x = 0 (y intercept)
Find f(x) = 0 (x intercepts)
Sketch

12. Assignment Lesson 4.4 Page 235
Exercises 5, 12, 19, 26, 33, 40, 47, 54, 61, 68, 73, 83, 91, 95, 96