Relative Extrema and More Analysis of Functions

1. Relative Extrema and More Analysis of Functions Section 5.2

2. Critical Points vs. Stationary Points • Critical Point – A point in the domain of f where f where f has a horizontal tangent line or is not differentiable • Stationary Point – A point on the graph where the

3. 1st Derivative Test • If extending left from and extending right from , then f has a relative maximum at • If extending left from and extending right from , then f has a relative maximum at • If has the same sign as it extend in either direction, then f does not have a relative extrema at

4. In other words….. • If a function increases from the left and decreases to the right of , then there is a relative maximum at the point • If a function decreases from the left and increases to the right of , then there is a relative minimum at the point • If the sign of the derivative does not change, then the function either increases or decreases and does not have a relative max or min

5. Increase: _________ Stationary Points:________ Decrease: _________ Inflection Points: ________ Concave Up: _________Rel. Maximum: _________ Concave Down: _______Rel. Minimum: _________

6. Increase: _________Stationary Points:________ Decrease: _________Inflection Points: ________ Concave Up: _________Rel. Maximum: _________ Concave Down: ________Rel. Minimum: _________

7. Examples with Graphs • Give the relative max, min, and inflection points 4 2 3

8. Practicepg. 287 (7, 9, 15 - 18, 27, 28, 31 – 34)