Lesson 4.1 Extreme Values


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Lesson 4.1 Extreme Values

  1. 1. 4.1 Extreme Values of Functions
  2. 2. Def : Absolute Extreme Values <ul><li>Let f be a function with domain D. Then f (c) is the… </li></ul><ul><li>a) Absolute (or global) maximum value on D iff f (x) ≤ f (c) for all x in D. </li></ul><ul><li>b) Absolute (or global) minimum value on D iff f (x) ≥ f (c) for all x in D. </li></ul><ul><li>* Note: The max or min VALUE refers to the y value. </li></ul>
  3. 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. 4. Thm 1 : Extreme Value Theorem <ul><li>If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval. </li></ul>* What is the “worst case scenario,” and what happens then?
  5. 5. Def : Local Extreme Values <ul><li>Let c be an interior point of the domain of f . Then f (c) is a… </li></ul><ul><li>a) Local (or relative) maximum value at c iff f (x) ≤ f (c) for all x in some open interval containing c. </li></ul><ul><li>b) Local (or relative) minimum value at c iff f (x) ≥ f (c) for all x in some open interval containing c. </li></ul>* A local extreme value is a max or min in a “neighborhood” of points surrounding c
  6. 6. Identify Absolute and Local Extrema * Local extremes are where f ’ (x)=0 or f ’ (x) DNE Abs & Loc Min Abs & Loc Max Loc Max Loc Min Loc Min
  7. 7. Thm 2 : Local Extreme Values <ul><li>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. </li></ul>
  8. 8. Def : Critical Point <ul><li>A critical point is a point in the interior of the domain of f at which either… </li></ul><ul><ul><ul><li>1) f ’(x)= 0 , or </li></ul></ul></ul><ul><ul><ul><li>2) f ’(x) does not exist </li></ul></ul></ul>
  9. 9. Finding Absolute Extrema on [a,b]… <ul><li>To find the absolute extrema of a continuous function f on [a,b]… </li></ul><ul><li>Evaluate f at the endpoints a and b. </li></ul><ul><li>Find the critical numbers of f on [a,b]. </li></ul><ul><li>Evaluate f at each critical number. </li></ul><ul><li>Compare values. The greatest is the absolute max and the least is the absolute min. </li></ul>
  10. 10. Check for Understanding… (True or False??) <ul><li>Absolute extrema can occur at either interior points or endpoints. </li></ul><ul><li>A function may fail to have a max or min value. </li></ul><ul><li>A continuous function on a closed interval must have an absolute max or min. </li></ul><ul><li>An absolute extremum is also a local extremum. </li></ul><ul><li>Not every critical point or end point signals the presence of an extreme value. </li></ul>