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4.3 Extreme Values of Functions

4.3 Extreme Values of Functions. Borax Mine, Boron, CA Photo by Vickie Kelly, 2004. Greg Kelly, Hanford High School, Richland, Washington. Local Extreme Values:. A local maximum is the maximum value within some open interval.

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4.3 Extreme Values of Functions

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  1. 4.3 Extreme Values of Functions Borax Mine, Boron, CA Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington

  2. Local Extreme Values: A local maximum is the maximum value within some open interval. A local minimum is the minimum value within some open interval.

  3. Local extremes are also called relative extremes. Absolute maximum (also local maximum) Local maximum Local minimum

  4. Notice that local extremes in the interior of the function occur where is zero or is undefined. Absolute maximum (also local maximum) Local maximum Local minimum

  5. 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 exists at c, then

  6. Critical Point: A point in the domain of a function f at which or does not exist is a critical point of f . Critical points where are called stationary points. Note: Maximum and minimum points in the interior of a function always occur at critical points, but critical points are not always maximum or minimum values.

  7. Critical points are not always extremes! (not an extreme)

  8. (not an extreme) p

  9. The First Derivative Test Let c be a critical point of a function f that is continuous on some open interval containing c. If f is differentiable on the interval (except possibly at c), then • If changes from negative to positive at c, • then f(c) is a relative minimum. 2. If changes from positive to negative at c, then f(c) is a relative maximum.

  10. Possible extreme at . Set Example: Use the first derivative test to find the relative extrema of: We can use a chart to organize our thoughts. First derivative test: negative positive positive

  11. Possible extreme at . Set maximum at minimum at Example: Graph Use the first derivative test to find the relative extrema of: First derivative test:

  12. There is a local maximum at (0,4) because for all x in and for all x in (0,2) . There is a local minimum at (2,0) because for all x in (0,2) and for all x in . NOTE: On the AP Exam, it is not sufficient to simply draw the chart and write the answer. You must give a written explanation! First derivative test:

  13. The Second Derivative Test (the easier way!!) If x = c is a critical point such that , and the second derivative exists on the interval containing c, then • If then f(c) is a relative minimum. 2. If then f(c) is a relative maximum. If , the test fails. In such cases you have to use the First Derivative Test.

  14. Possible extreme at . Because the second derivative at x =0 is negative, the graph is concave down and therefore (0,4) is a local maximum. Because the second derivative at x =2 is positive, the graph is concave up and therefore (2,0) is a local minimum. Example: Graph Use the second derivative test to find the relative extrema of:

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