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

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.

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

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

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

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

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 _____________________.

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

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

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