2. §3.4 Concavity Concave Up Concave Down

3. Inflection Points Concavity Changes Concave Up Concave Down

4. To Determine Concavity f’(x) > 0 means Increasing f’(x) < 0 means Decreasing f’(x) = 0means Constant f”(x) > 0means Concave Up f”(x) < 0means Concave Down f”(x) = 0means Inflection Point

5. Steps to follow: Find critical values. f’(x) = 0 & f”(x) = 0 Set-up intervals & tests Write conclusions (Use Chart)

6. Intervalf(x)f’(x)f”(x)Conclusion -∞<x<c 1 X +-0+-0 +-0+-0 Inc or Dec U or ∩ x = c 1 f( c 1 ) = # f’( c 1 ) = # f”( c 1 ) = # Rel. Extrema (Max./Min.) Inflection Point c 1 <x<c 2 X +-0+-0 +-0+-0 Inc or Dec U or ∩ x = c 2 f( c 2 ) = # f’( c 2 ) = # Rel. Extrema (Max./Min.) Inflection Point c 2 <x< ∞ X +-0+-0 +-0+-0 Inc or Dec U or ∩

7. New Test for Rel. Extrema! If f”(x) >0, then Relative Minimum. -Valley = low point If f”(x) < 0, then Relative Maximum. -Peak = high point

8. When f”(x) = 0… Use the first derivative test!