1. CONCAVITY AND SECOND DERIVATIVE RIZZI – CALC BC

2. WARM UP Given derivative graph below, find a. intervals where the original function is increasing b. intervals where the original function is decreasing c. x-coordinates of the local maximums and minimums of the function

3. WHAT IS CONCAVITY? Concavity is another physical interpretation of a function

4. SECOND DERIVATIVE = CONCAVITY The second derivative tells us intervals where the function is concave up and concave down.

5. INFLECTION POINTS Inflection points are the points where the graph changes concavity

6. INTERVALS OF CONCAVE UP AND DOWN Find the intervals of concavity for the function Step 1: Find the second derivative, f”(x) = 0 Step 2: Determine the x-coordinates of the points of inflection Step 3: Test the concavity for each interval in f”(x)

8. AP PROBLEM

9. SECOND DERIVATIVE TEST FOR EXTREMA

10. TRY IT – SECOND DERIVATIVE TEST Find the relative extrema of the function. Step 1: Find the critical numbers of the function, where f’(x) = 0 Step 2: Find the second derivative and test each x-value to see the concavity at each point. Step 3: Plug x-values into original to find coordinates

12. COMPARISON OF 1 ST AND 2 ND DERIVATIVE =0IntervalsExtrema Test 1 st DerivativeCritical Points (m=0) Increasing/ Decreasing Use critical points and intervals of increasing/decreasing 2 nd DerivativeInflection Points (concavity changes) Concave Up/DownUse critical points and concavity