Concavity at limits

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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:)
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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)
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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]
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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]
 
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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]
 
rachmaninoff
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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