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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