Caculus: Critical Numbers

  1. 1. The problem statement, all variables and given/known data
    The first derivative of the function f is given by f '(x)= (((cos^2)x)/(x)) - 1/5
    How many critical numbers does f have on the open interval (0,10)?


    2. Relevant equations



    3. The attempt at a solution
    I already got this question wrong but I don't know why I got it wrong. The answer is 3 but when I graph it I see 6 critical numbers. So why is it 3 and not 6? Please explain.
     
  2. jcsd
  3. Let's clear up any confusion about what the problem is ...

    [tex]f'(x)=\frac{\cos^2 x}{x}-\frac 1 5[/tex]

    Correct?
     
  4. Correct
     
  5. I tried taking the derivative of the derivative and graphed that but it still gives me an image of 6 critical points.
     
  6. HallsofIvy

    HallsofIvy 40,939
    Staff Emeritus
    Science Advisor

    I don't understand why that would happen. A critical point occurs where the derivative is 0 or does not exist. Clearly your derivative does not exist at x= 0. To determine where f'= 0, I graphed y= 5cos2(x) and y= x. They cross in 3 points.

    Wait, did you differentiate again? You said what you gave was f '. To determine where f has critical points, you should be graphing that, not its derivative.
     
    Last edited: Jan 25, 2008
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