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Concavity at limits

  1. Jul 19, 2005 #1
    Just four questions here :biggrin: :

    1) For a function f(x), [itex] \exists f''\left( x \right) [/itex] for [itex] \left\{ {x|\left( { - \infty ,a} \right) \cup \left( {a,\infty } \right)} \right\} [/itex], and [tex] \mathop {\lim }\limits_{x \to a} f\left( x \right) = \infty [/tex].
    Then, is it true that
    [tex] \mathop {\lim }\limits_{x \to a} f''\left( x \right) > 0 \, {?} [/tex]

    (...in the sense that always [itex] \exists \, \varepsilon > 0 [/itex] such that [itex] \forall x \in \left[ {a - \varepsilon ,a + \varepsilon } \right] [/itex] where [itex] x \ne a [/itex], [itex] f''\left( {x} \right) > 0 [/itex], that is :smile:)
    --------------------------------------------------------------------------
    2) And, if
    [tex] \mathop {\lim }\limits_{x \to a} f\left( x \right) = - \infty [/tex], then
    [tex] \mathop {\lim }\limits_{x \to a} f''\left( x \right) < 0 \, {?} [/tex], right?

    If both statements are true, what's the name of the theorem stating them?
    (or explaining them, I suppose)
    ---------------------------------------------------------------------------
    3) Now, let [itex] f^{\left( n \right)} \left( x \right) [/itex] represent the n'th derivative of f(x). If [tex] \mathop {\lim }\limits_{x \to a} f\left( x \right) = \infty [/tex],
    is it true that if [tex] \exists f^{\left( n \right)} \left( x \right) [/tex],
    then [tex] \mathop {\lim }\limits_{x \to a} f^{\left( n \right)} \left( x \right) > 0 \, {?}[/tex]
    --------------------------------------------------------========
    4) Finally, if [tex] \mathop {\lim }\limits_{x \to a} f\left( x \right) = \infty [/tex],
    is it true that if [tex] \exists f^{\left( n \right)} \left( x \right) [/tex],
    then [tex] \mathop {\lim }\limits_{x \to a} f^{\left( n \right)} \left( x \right) = \infty \, {?}[/tex]
     
    Last edited: Jul 20, 2005
  2. jcsd
  3. Jul 20, 2005 #2
    1) This need not be true. For example:

    [tex]f(x) = \frac{\sin \left(\frac{1}{x}\right)+2}{x^2}[/tex]
    It is two times differentiable everywhere except at the origin. And
    [tex] \mathop {\lim }\limits_{x \to 0} f\left( x \right) = \infty [/tex], but [tex]f''(x)[/tex] (and any other of it's derivatives) is alternatingly positive and negative when you approach the origin.

    2) Counterexample: take [tex]g(x)=-f(x)[/tex]

    3,4) Counterexample: again [tex]f[/tex]
     
  4. Jul 20, 2005 #3
    However, replace [itex]f^{(n)}(x)[/itex] with [itex] | f^{(n)}(x)| [/itex] and (3) (and thus (1)) works. For if the nth derivative approaches zero, the higher derivatives ( >n ) also approach zero, and what you have is a finite limit. You can prove this with the definition of the derivative, and an epsilon/delta argument.
     
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