1. 4.1 Extreme Values on Functions

2. Vocab/Formulas Extreme Values: Absolute/Global Extrema:
Absolute Max: f(c) iff f(x)≤f(c) for all x
Absolute Min: f(c) iff f(x)≥f(c) for all x
Local/Relative Extrema:
Local Max: f(c) iff f(x)≤f(c) for all x in some open interval containing c
Local Min: f(c) iff f(x)≥f(c) for all x in some open interval containing c

4. Experiment: Where do max’s and min’s happen?

5. Local Extreme Values Theorem
Extreme Value Theorem
If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval.
Local Extreme Values Theorem
If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f ’ exists at c, then f ’(c)=0

6. EXTREME VALUES OCCUR ONLY AT CRITICAL POINTS AND ENDPOINTS
Vocab/Formulas
Critical Point: A point on the interior of a function f at which f ’ = 0 or f ’ DNE
EXTREME VALUES OCCUR ONLY AT CRITICAL POINTS AND ENDPOINTS

7. Example 1: Find the extreme values for the given graphs.

8. Example 2: Find the extreme values and where they occur for the function on the given interval.
Graphically
Analytically

9. Example 3: Find the extreme values and where they occur for the function on the given interval.
Graphically
Analytically

10. Example 4: Find the extreme values and where they occur for the function.
Graphically
Analytically

11. Assignment
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