1. Concave Upward, Concave Downward
Concavity
and the
2nd Derivative Test
Objectives:
Be able to determine where a function is concave upward or concave downward with the use of calculus.
Be able to apply the second derivative test to find the relative extrema of a function.
Critical Vocabulary:
Concave Upward, Concave Downward
Warm Up: Find the 2nd derivative of each function

2. Let f be differentiable on a open interval I
Let f be differentiable on a open interval I. The graph of f is _____________________ on I if f’(x) is increasing on the interval and concave __________________ on I if f’(x) is decreasing on the interval.

3. These is called a point of _______. Concavity changes at these points.

4. 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 f is ______________ in I
2. If f’’(x) < 0 for all x in I, then f is ______________ in I

5. Example 1: Find the open intervals on which the following function
is concave upward or concave downward.
Concave Up:
Concave Down:
Interval
Test Value
Sign of f’’(x)
Conclusion

6. Example 2: Find the open intervals on which the following function
is concave upward or concave downward.
Concave Up:
Concave Down:
Interval
Test Value
Sign of f’’(x)
Conclusion

7. Let f be a function such that f’(c) = 0 and the second derivative of f exists on a open interval containing c.
If f’’(c) > 0, then f(c) is a ____________________ of f.
2. If f’’(c) < 0, then f(c) is a ____________________ of f.
3. If f’’(c) = 0, then the __________. In such case, you use the _____________________.

8. Example 3: Find the relative extrema for f(x) = -3x5 + 5x3 using
the second derivative test.
Point
Sign f’’(x)
Conclusion

9. Example 4: Find the relative extrema for using
the second derivative test.
Point
Sign f’’(x)
Conclusion

10. Page 342 #1-29 odd, 33, 45, 47
(MUST USE CALCULUS!!!!)