Concavity at Limits: 4 Questions

[bomba923]

Just four questions here :

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

(...in the sense that always \exists \, \varepsilon > 0 such that \forall x \in \left[ {a - \varepsilon ,a + \varepsilon } \right] where x \ne a, f''\left( {x} \right) > 0, that is )
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2) And, if
\mathop {\lim }\limits_{x \to a} f\left( x \right) = - \infty, then
\mathop {\lim }\limits_{x \to a} f''\left( x \right) < 0 \, {?}, 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 f^{\left( n \right)} \left( x \right) represent the n'th derivative of f(x). If \mathop {\lim }\limits_{x \to a} f\left( x \right) = \infty,
is it true that if \exists f^{\left( n \right)} \left( x \right),
then \mathop {\lim }\limits_{x \to a} f^{\left( n \right)} \left( x \right) > 0 \, {?}
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4) Finally, if \mathop {\lim }\limits_{x \to a} f\left( x \right) = \infty,
is it true that if \exists f^{\left( n \right)} \left( x \right),
then \mathop {\lim }\limits_{x \to a} f^{\left( n \right)} \left( x \right) = \infty \, {?}

 

[Timbuqtu]

1) This need not be true. For example:

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

2) Counterexample: take g(x)=-f(x)

3,4) Counterexample: again f

 

[rachmaninoff]

However, replace f^{(n)}(x) with | f^{(n)}(x)| 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.