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Find intervals of f increasing/decreasing

  1. Jul 9, 2010 #1
    1. The problem statement, all variables and given/known data

    (a) Find the intervals on which f is increasing or decreasing.
    (b) Find the local maximum and minimum values of f.
    (c) Find the intervals of concavity and the inflection points.

    x2/(x2+3)

    2. Relevant equations



    3. The attempt at a solution

    a) I find that
    f'(x)=6x/(x2+3)2

    I am stuck because (x2+3)2 has no solutions and I don't know how to define x in order to proceed further.
    If anybody can help me out understanding this, I will be very thankful.
     
  2. jcsd
  3. Jul 9, 2010 #2

    hunt_mat

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    Homework Helper

    Look at the function to see what it's it does. as [tex]x\rightarrow\pm\infty[/tex], [tex]f(x)\rightarrow 1[/tex]. Also note that f(x) is an even function. Your calculation shows that the one and only minimum is at x=0. This should allow you to answer the question.

    Mat
     
  4. Jul 9, 2010 #3

    Mark44

    Staff: Mentor

    Why are you just looking at the denominator? Since x2+3 has no real solutions, this means that the domain of f is the entire real line.

    f is increasing on intervals for which f'(x) > 0, and is decreasing on intervals for which f'(x) < 0.

    You're going to need the second derivative, too, in this problem.
     
  5. Jul 9, 2010 #4
    1) How do you know it is an even function?
    2) Why [tex]f(x)\rightarrow 1[/tex]?
     
  6. Jul 9, 2010 #5

    Mark44

    Staff: Mentor

    1) Because f(-x) = f(x) for all x
    2) Because x2/(x2 + 3) = 1 - 3/(x2 + 3). As x --> inf, f(x) --> 1. As x --> -int, f(x) --> 1.
     
  7. Jul 11, 2010 #6
    So how can I find the intervals of concavity based on a second derivative?
     
  8. Jul 11, 2010 #7

    Dick

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    Science Advisor
    Homework Helper

    Take the second derivative and figure out where it's positive and where it's negative?
     
  9. Jul 11, 2010 #8
    I did so, and in this case, x=1. however, the graph is concave down at 0 and it has nothing to do with a point 1.
     
  10. Jul 12, 2010 #9

    Dick

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    Science Advisor
    Homework Helper

    x=1 isn't an 'interval of concavity'. It's just a point. If you mean it's an inflection point, yes it is. But it's not the only one.
     
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