![Thm 1: 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.
* What is the “worst case scenario,”
and what happens then?](https://image.slidesharecdn.com/lecture11relativeextrema-200523001236/85/Lecture-11-relative-extrema-4-320.jpg)
![Finding Absolute
Extrema on [a,b]…
To find the absolute extrema of a
continuous function f on [a,b]…
1. Evaluate f at the endpoints a and b.
2. Find the critical numbers of f on [a,b].
3. Evaluate f at each critical number.
4. Compare values. The greatest is the
absolute max and the least is the
absolute min.](https://image.slidesharecdn.com/lecture11relativeextrema-200523001236/85/Lecture-11-relative-extrema-9-320.jpg)
Lecture 11(relative extrema)
- 2. Def: Absolute Extreme Values Let f be a function with domain D. Then f(c) is the… a) Absolute (or global) maximum value on D iff f(x) ≤ f(c) for all x in D. b) Absolute (or global) minimum value on D iff f(x) ≥ f(c) for all x in D. * Note: The max or min VALUE refers to the y value.
- 3. Where can absolute extrema occur? * Extrema occur at endpoints or interior points in an interval. * A function does not have to have a max or min on an interval. * Continuity affects the existence of extrema. Min Max Min No Max No Max No Min
- 4. Thm 1: 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. * What is the “worst case scenario,” and what happens then?
- 5. Def: Local Extreme Values Let c be an interior point of the domain of f. Then f(c) is a… a) Local (or relative) maximum value at c iff f(x) ≤ f(c) for all x in some open interval containing c. b) Local (or relative) minimum value at c iff f(x) ≥ f(c) for all x in some open interval containing c. * A local extreme value is a max or min in a “neighborhood” of points surrounding c
- 6. Identify Absolute and Local Extrema Abs & Loc Min Abs & Loc Max Loc Max Loc Min Loc Min * Local extremes are where f ’(x)=0 or f ’(x) DNE
- 7. Thm 2: Local Extreme Values 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.
- 8. Def: Critical Point A critical point is a point in the interior of the domain of f at which either… 1) f ’(x)= 0, or 2) f ’(x) does not exist
- 9. Finding Absolute Extrema on [a,b]… To find the absolute extrema of a continuous function f on [a,b]… 1. Evaluate f at the endpoints a and b. 2. Find the critical numbers of f on [a,b]. 3. Evaluate f at each critical number. 4. Compare values. The greatest is the absolute max and the least is the absolute min.
- 10. Check for Understanding… (True or False??) 1. Absolute extrema can occur at either interior points or endpoints. 2. A function may fail to have a max or min value. 3. A continuous function on a closed interval must have an absolute max or min. 4. An absolute extremum is also a local extremum. 5. Not every critical point or end point signals the presence of an extreme value.