Local Extrema, Absolute Extrema, Increasing, Decreasing ...
Calculus BC Chapter 4.1,4.3 Notes
Definition: Local Extreme Values (Pg. 179)
a) local maximum value
b) local minimum value
Definition: Absolute (Global) Extreme Values (Pg. 177)
a) absolute maximum value
b) absolute minimum value
Definition: Critical Point (Pg. 180)
Definition: Increasing Function, Decreasing Function (Pg. 188)
Definition (Corollary): Constant (Pg. 189)
Implications:
If a function f has a local extremum at a number c in an open interval
If a function f has a local extremum at a number c on a closed interval
Let c be in the domain of f: [pic] or [pic]does not exist
A function is increasing on an open interval
A function is decreasing on an open interval
A function is constant on an open interval
First Derivative Test (applies to a continuous function f(x)):
At a critical point c:
1. If [pic]changes sign from positive to negative at c, then f has a local maximum value at c.
2. If [pic]changes sign from negative to positive at c, then f has a local minimum value at c.
3. If [pic]does not change sign at c, then f has no local extreme value at c.
(Left Endpoint a) If [pic], then f has a local maximum at a.
If [pic], then f has a local minimum at a.
(Right Endpoint b) If [pic], then f has a local maximum at b.
If [pic], then f has a local minimum at b.
Using the graph of the derivative to find critical points, increasing/decreasing behavior, and the values of x where the function has local extremes
Use the first derivative test to find the critical points, local extremes, as well as the absolute extremes.
1. [pic]
[pic]
[pic]
2. [pic]
[pic]
[pic]
3. [pic]
[pic]
[pic]
4. [pic]
[pic]
[pic]
Homework.
1. [pic]
Find the local extrema of f and the intervals on which f is increasing or is decreasing, and sketch the graph of f.
2. [pic]
Find the local extrema of f (and classify) and the intervals on which f is increasing or is decreasing, and sketch the graph of f.
3. [pic]
Find the local extrema of f (and classify) on [pic] and the subintervals on which f is increasing or is decreasing. Sketch the graph of f.
4. [pic] on [-1,1]
Graph [pic]on the indicated interval. Estimate the x-coordinates of the local extrema of f and classify each local extrema.
5. Sketch the graph of a continuous function f that satisfies the given conditions.
[pic]is undefined;
[pic]
[pic]
[pic]
6. Sketch the graph of a continuous function f that satisfies the given conditions.
[pic]
[pic]
[pic]
[pic]
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