Suppose that $f$, $f'$ and $f''$ are positive on $mathbb{R}$ then $lim_{xtoinfty } f(x) = infty$.
$begingroup$
I have been trying to prove this for a quite a long time but I have been unsuccessful so far.
We need to show that for any $M>0$ there exists a $d>0$ such that $f(x)>M$ for all $x>d$ in order to show that $lim_{xtoinfty } f(x) = infty$.
Here's how I try to begin the proof:
Let $M>0$ be given. Let $g(x):=f(x)-M$. We need to show that there is a point $d>0$ such that $g(x)>0$ for all $x>d$. Also we have been given that $g'(x)>0$ and $g''(x)>0$ for all $xinmathbb{R}$. Thus, it must be that $g$ and $g'$ are strictly increasing.
I'm not sure how to proceed after that. I tried a lot using MVT but I couldn't succeed. Hints would be appreciated.
real-analysis
asked Nov 29 '18 at 16:12
Ashish KAshish K
831613
$endgroup$
add a comment |
$begingroup$
I have been trying to prove this for a quite a long time but I have been unsuccessful so far.
We need to show that for any $M>0$ there exists a $d>0$ such that $f(x)>M$ for all $x>d$ in order to show that $lim_{xtoinfty } f(x) = infty$.
Here's how I try to begin the proof:
Let $M>0$ be given. Let $g(x):=f(x)-M$. We need to show that there is a point $d>0$ such that $g(x)>0$ for all $x>d$. Also we have been given that $g'(x)>0$ and $g''(x)>0$ for all $xinmathbb{R}$. Thus, it must be that $g$ and $g'$ are strictly increasing.
I'm not sure how to proceed after that. I tried a lot using MVT but I couldn't succeed. Hints would be appreciated.
real-analysis
asked Nov 29 '18 at 16:12
Ashish KAshish K
831613
$endgroup$
add a comment |
$begingroup$
I have been trying to prove this for a quite a long time but I have been unsuccessful so far.
We need to show that for any $M>0$ there exists a $d>0$ such that $f(x)>M$ for all $x>d$ in order to show that $lim_{xtoinfty } f(x) = infty$.
Here's how I try to begin the proof:
Let $M>0$ be given. Let $g(x):=f(x)-M$. We need to show that there is a point $d>0$ such that $g(x)>0$ for all $x>d$. Also we have been given that $g'(x)>0$ and $g''(x)>0$ for all $xinmathbb{R}$. Thus, it must be that $g$ and $g'$ are strictly increasing.
I'm not sure how to proceed after that. I tried a lot using MVT but I couldn't succeed. Hints would be appreciated.
real-analysis
asked Nov 29 '18 at 16:12
Ashish KAshish K
831613
$endgroup$
I have been trying to prove this for a quite a long time but I have been unsuccessful so far.
We need to show that for any $M>0$ there exists a $d>0$ such that $f(x)>M$ for all $x>d$ in order to show that $lim_{xtoinfty } f(x) = infty$.
Here's how I try to begin the proof:
Let $M>0$ be given. Let $g(x):=f(x)-M$. We need to show that there is a point $d>0$ such that $g(x)>0$ for all $x>d$. Also we have been given that $g'(x)>0$ and $g''(x)>0$ for all $xinmathbb{R}$. Thus, it must be that $g$ and $g'$ are strictly increasing.
I'm not sure how to proceed after that. I tried a lot using MVT but I couldn't succeed. Hints would be appreciated.
real-analysis
real-analysis
asked Nov 29 '18 at 16:12
Ashish KAshish K
831613
asked Nov 29 '18 at 16:12
Ashish KAshish K
831613
asked Nov 29 '18 at 16:12
Ashish KAshish K
831613
asked Nov 29 '18 at 16:12
Ashish KAshish K
831613
asked Nov 29 '18 at 16:12
Ashish KAshish K
831613
831613
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint:
The function is positive, increasing and convex.
For all $x > c$ we have $f(x) > f(c) + f'(c)(x-c)$
answered Nov 29 '18 at 16:19
RRLRRL
49.5k42573
$endgroup$
$begingroup$
Thanks for the hint. That helped. I completed my proof!
$endgroup$
– Ashish K
Nov 29 '18 at 16:37
$begingroup$
@AshishK: You're welcome. Good work.
$endgroup$
– RRL
Nov 29 '18 at 16:55
add a comment |
Your Answer
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1 Answer
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active
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1 Answer
1
active
oldest
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active
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active
oldest
votes
$begingroup$
Hint:
The function is positive, increasing and convex.
For all $x > c$ we have $f(x) > f(c) + f'(c)(x-c)$
answered Nov 29 '18 at 16:19
RRLRRL
49.5k42573
$endgroup$
$begingroup$
Thanks for the hint. That helped. I completed my proof!
$endgroup$
– Ashish K
Nov 29 '18 at 16:37
$begingroup$
@AshishK: You're welcome. Good work.
$endgroup$
– RRL
Nov 29 '18 at 16:55
add a comment |
$begingroup$
Hint:
The function is positive, increasing and convex.
For all $x > c$ we have $f(x) > f(c) + f'(c)(x-c)$
answered Nov 29 '18 at 16:19
RRLRRL
49.5k42573
$endgroup$
$begingroup$
Thanks for the hint. That helped. I completed my proof!
$endgroup$
– Ashish K
Nov 29 '18 at 16:37
$begingroup$
@AshishK: You're welcome. Good work.
$endgroup$
– RRL
Nov 29 '18 at 16:55
add a comment |
$begingroup$
Hint:
The function is positive, increasing and convex.
For all $x > c$ we have $f(x) > f(c) + f'(c)(x-c)$
answered Nov 29 '18 at 16:19
RRLRRL
49.5k42573
$endgroup$
Hint:
The function is positive, increasing and convex.
For all $x > c$ we have $f(x) > f(c) + f'(c)(x-c)$
answered Nov 29 '18 at 16:19
RRLRRL
49.5k42573
answered Nov 29 '18 at 16:19
RRLRRL
49.5k42573
answered Nov 29 '18 at 16:19
RRLRRL
49.5k42573
answered Nov 29 '18 at 16:19
RRLRRL
49.5k42573
49.5k42573
$begingroup$
Thanks for the hint. That helped. I completed my proof!
$endgroup$
– Ashish K
Nov 29 '18 at 16:37
$begingroup$
@AshishK: You're welcome. Good work.
$endgroup$
– RRL
Nov 29 '18 at 16:55
add a comment |
$begingroup$
Thanks for the hint. That helped. I completed my proof!
$endgroup$
– Ashish K
Nov 29 '18 at 16:37
$begingroup$
@AshishK: You're welcome. Good work.
$endgroup$
– RRL
Nov 29 '18 at 16:55
$begingroup$
Thanks for the hint. That helped. I completed my proof!
$endgroup$
– Ashish K
Nov 29 '18 at 16:37
$begingroup$
Thanks for the hint. That helped. I completed my proof!
$endgroup$
– Ashish K
Nov 29 '18 at 16:37
$begingroup$
@AshishK: You're welcome. Good work.
$endgroup$
– RRL
Nov 29 '18 at 16:55
$begingroup$
@AshishK: You're welcome. Good work.
$endgroup$
– RRL
Nov 29 '18 at 16:55
add a comment |
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