3.1 extreme values of functions

3.1 Extreme Values

of Function

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3.1 Extreme Values

of Function

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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.

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

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?

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

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

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.

Def: Critical Point

A critical point is a point in the interior of the domain of f at which either…

1) f ’(x)= 0, or2) f ’(x) does not exist

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.

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.