1. AP Calculus Section 3.4 Concavity and the Second Derivative Test
Student Notes
Definitions, Theorems, and Examples

2. Definition
Let f be differentiable on an open interval I. The graph of f is concave upward on I if f’ is increasing on the interval and concave downward on I if f’ is decreasing on the interval.
Concave downward. f’ is decreasing. The graph of f lies below its tangent lines.
Concave upward. f’ is increasing. The graph of f lies above its tangent lines.

3. The graph of a function f is concave upward at the point (c, f(c)) if f(c) exists and if for all x in some open interval containing c, the point (x, f(x)) on the graph of f lies above the corresponding point on the graph of the tangent line to f at c.
Imagine holding a ruler along the tangent line through the point (c, f(c)): if the ruler supports the graph of f near (c, f(c)), then the graph of the function is concave upward.
The graph of a function f is concave downward at the point (c,f(c)) if f(c) exists and if for all x in some open interval containing c, the point (x, f(x)) on the graph of f lies below the corresponding point on the graph of the tangent line to f at c.
In this situation, the graph of f supports the ruler.

4. You try:

5. Test for Concavity Theorem 3.7
Let f be a function whose second derivative exists on an open interval I.
1.) If f’’(x) > 0 for all x in I, then the graph of f is concave upward in I.
2.) If f’’(x) < 0 for all x in I, then the graph of f is concave downward in I.
To apply Theorem 3.7, locate the x-values at which f’’(x) = 0 or f’’ does not exist. Then, use these x-values to determine test intervals. Finally, test the sign of f’’(x) in each of the test intervals.

6. A straight line is neither concave upward nor concave downward
A straight line is neither concave upward nor concave downward. Can you explain why???

7. Example 1 (#3)
Determine the open intervals on which 𝑓 𝑥 = 24 𝑥 is concave upward or concave downward.
Interval
Test Value
Sign of f’’(x)
Conclusion

9. Example 2 (#10)
Determine the open intervals on which y =𝑥+ 2 𝑠𝑖𝑛𝑥 on (−𝜋, 𝜋) is concave upward or concave downward.
Interval
Test Value
Sign of f’’(x)
Conclusion

11. Definition
Let f be a function that is continuous on an open interval and let c be a point in the interval. If the graph of f has a tangent line at this point (c, f(c)), then this point is a point of inflection of the graph of f if the concavity of f changes from upward to downward (or downward to upward) at the point.

12. The graph of a function crosses its tangent line at a point of inflection.

13. Example 3 A (#16)
Determine the points of inflection and discuss the concavity of the graph of 𝑓 𝑥 = 𝑥 3 (𝑥−4).
Interval
Test Value
Sign of f’’(x)
Conclusion

15. Example 3 B (#24)
Determine the points of inflection and discuss the concavity of the graph of 𝑓 𝑥 =𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥 on [0, 2𝜋].
Interval
Test Value
Sign of f’’(x)
Conclusion

17. Theorem 3.9 Second Derivative Test
Let f be a function such that f’(c)=0 and the second derivative of f exists on an open interval containing c.
1.) If f’’(c) > 0, then f has a relative minimum at (c, f(c)).
2.) If f’’(c) < 0, then f has a relative maximum at (c, f(c)).
If f’’(c) = 0, the test fails. f may have a relative maximum, a relative minimum, or neither. In these cases, you can use the First Derivative Test.

18. Example 4 A (#34)
Use the Second Derivative Test to find all relative extrema of
𝑔 𝑥 =− 1 8 (𝑥+2 ) 2 (𝑥−4 ) 2 .

20. Example 4 b (#40)
Use the Second Derivative Test to find all relative extrema of
𝑓 𝑥 =2𝑠𝑖𝑛𝑥+𝑐𝑜𝑠2𝑥 on [0, 2𝜋].

22. Assignment – Section 3.4 p. 195-197 #5, 9, 15, 19, 26, 35, 37, 67 or 68 (you choose)