If $f'(x)$ tends to $\infty$ then $f(x)$ does the same

Question

I noticed that when $f'(x)$ tends to $+\infty$ as $x$ tends to $+\infty$, then $f(x)$ must tend to $+\infty$ as $x$ tends to $+\infty$ as well. I'm stuck at the proof though.

If you implement the mean value theorem for $f(x)$ in $(x,x+1)$ you'll get:

$f(x+1) - f(x)= f'(b)$ where $b>x$

now if u take limits for $x\rightarrow+\infty$, b must tend to $+\infty$ as well. So, $\lim_{x\to\infty}$ $f(x+1) - f(x)$ = $+\infty$, as $x$ tends to $+\infty$

Also, $f(x)$ must be increasing as $x$ tends to $+\infty$. Can somebody continue this or show me another way of proving this?

Did · Answer 1 · 2013-06-12T13:09:49.930

12

Since $f'(x)\to+\infty$ when $x\to+\infty$, there exists $x_0$ such that $f'(x)\geqslant1$ for every $x\geqslant x_0$. Thus, $f(x)\geqslant x-x_0+f(x_0)$ for every $x\geqslant x_0$. Since $x-x_0+f(x_0)\to+\infty$ when $x\to+\infty$, this implies that $f(x)\to+\infty$ when $x\to+\infty$.

edited Jun 12 '13 at 13:09

answered Jun 12 '13 at 12:28

Did

279,727

2

did u just think of that? ._. – Plom Jun 12 '13 at 12:34
@Plom This comes from the interpretation of $f'$ as the "gradient" of the graph: The graph gets steeper, so clearly it lies above a line with gradient 1 when $x$ is large enough. – Jun 12 '13 at 12:50
Plom: Are you able to expand on the "Thus" step? – Did Jun 12 '13 at 13:09
@Did What justifies the inequality following the word "Thus,"? – Jack M Jun 12 '13 at 13:15
@JackM Surely you can answer to that... :-) – Did Jun 12 '13 at 13:49
(This is odd, I am nearly certain this answer was accepted at a time--but no more.) – Did Jun 12 '13 at 13:50
do u mean the mean value theorem part? or the f'(x)>=1 ? the latter sounds legit :/ – Plom Jun 12 '13 at 15:34
"Sounds legit" $\ne$ "is a full proof". – Did Jun 12 '13 at 15:35
i cant prove it from looking at it right now, but i have a hunch it can be proved with the definition of limits ._. ...just a hunch tho – Plom Jun 12 '13 at 15:37

If $f'(x)$ tends to $\infty$ then $f(x)$ does the same

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