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 )
--------------------------------------------------------------------------
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)
---------------------------------------------------------------------------
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 $$\mathop {\lim }\limits_{x \to a} f^{\left( n \right)} \left( x \right) > 0 \, {?}$$
--------------------------------------------------------========
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.