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  1. 1. 2.5 Critical Numbers Today in class we learned about critical numbers. The squares/rectangles indicate the local maximums and the stars indicate the relative maximums. It is called local because there is no maximum or minimum but relatively there are n number of maximums and minimums. In this case there are three maximums and three minimums. A local/relative maximum for  a function of f occurs at a  point x=a, if f(a) is the  largest   value of f in some interval  centered at x=a local/relative minimum ­­­>  smallest 1
  2. 2. Next, we learned how to draw f'(x). where f is decreasing on where f is increasing on (a,b), f'(x)<0 (a,b), f'(x) > 0 at the local extreme values f'(x)=0 2
  3. 3. A number C in the interior of the domain of a function f, is called a critical number if either: f'(c)=0 or f'(c)= d.n.e [does not exist] Determine the local extreme values. dec dec ng rea easi r sin eas incr g ing local minimum of f local maximum of f For this example we found the local extreme values by substituting 0 in for dy/dx. We found two values 0 and 2 to be the critical points. We then used a number line to find out where negative and positive outcomes occured. From our information, we were able to draw the resulting graph. 3
  4. 4. *Note* All critical points are not local extreme values. This is not a local extremum. It is called a point of inflection . This is an example of where the critical point is not a local extreme value. Again, we solved for our critcal point by substituting zero into the equation and used a number line to find negative and/or positive outcomes. Then we drew the graphs of f'(x) and f(x). 4
  5. 5. 5
  6. 6. So, basically we learned about critical numbers/points and how to draw the derivative function. We also went over question one in our homework due tomorrow. Homework is 1­20 odds + 22. Robert 6

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