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Maximum and Minimum Question?

  1. Oct 24, 2004 #1
    I'm frustrated beyond belief with a proof.

    Suppose we have an continuous even function with a domain of all real numbers. Now this function has limit as x goes to negative infinty equal to l and the limit as x goes to positive infinty is also equal to l.

    I want to show that this function will either have a maximum or a minimum.

    I'm not sure at all how to show this rigorously since I don't know how to apply the definition of a limit to limits at infinity. I think it has to do with bounds. And I need to do this without first derivative test.
  2. jcsd
  3. Oct 24, 2004 #2


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    Aren't the specifications you give consistent with the constant function f(x) = 1?
  4. Oct 24, 2004 #3
    Did you think my l was a 1. Maybe I should write it differently.

    lim (x-> -oo) f(x) = lim (x-> oo) f(x) = a
    and a is even.

    Want to show f has either a minimum or a maximum.
  5. Oct 24, 2004 #4


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    intuitively this makes sense, but rigorously you could show that unless it is a consant function (for example y = 1) then there must be a point where it switches between a positive and negative slope. Im not entirely sure what level of "rigorousness" you want.
  6. Oct 25, 2004 #5


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    First, you still have the problem that was pointed out by both selfadjoint and T@p:
    The constant function f(x)= a satisfies your conditions but does not have a maximum or minimum so the "theorem" as you stated it is not true.

    If f(x) is NOT A CONSTANT FUNCTION, then there exist some x0 such that f(x0) is not equal to a and so is either larger than or less than a.

    Assume f(x0)> a. Since limit as x-> infinity f(x)= a, there exist some x1> x0 such that f(x1)< f(x0).
    Similarly, since limit as x-> -infinity f(x)= a, there exist some x2< x0 such that f(x2)< f(x0). Since f is continuous on the closed and bounded interval [x2, x1] it must have both maximum and minimum values there. Now show that f has a maximum on -infinity to infinity.
  7. Oct 25, 2004 #6
    I need help. Find the minimum of y = Absolute value of (sinx + cosx + tanx + cotx +
    secx + cscx) Thanks Ruth Jackson the_perfect_mom@hotmail.com
  8. Oct 25, 2004 #7


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    I think you need to add something about x1 here. Choose it in such a way that if x>x1 then f(x)<f(x0). Similar change with how you select x2.
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