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 June 17th, 2017, 03:45 PM #1 Member   Joined: May 2017 From: France Posts: 57 Thanks: 1 A strange Inequality Hi, Let $\displaystyle g\in C^1([0,1],\mathbb R)$ Is it true that : $\displaystyle 4\times (\max(g)-\min(g))\geq \min(g')$ ? Cordially.
 June 27th, 2017, 05:52 AM #2 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100 It seems to work if you sketch out a few representative curves. From there you could probably develop a proof if you're interested beyond the "gotcha" phase.
 June 27th, 2017, 05:57 AM #3 Member   Joined: May 2017 From: France Posts: 57 Thanks: 1 Gotcha
June 27th, 2017, 06:48 AM   #4
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Quote:
 Originally Posted by Dattier Gotcha
You forgot "Cordially"

 June 27th, 2017, 06:51 AM #5 Member   Joined: May 2017 From: France Posts: 57 Thanks: 1 Gotcha
 June 27th, 2017, 07:00 AM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,327 Thanks: 2451 Math Focus: Mainly analysis and algebra I may have misunderstood, but $$g:[0,1]\mapsto \mathbb R \quad \text{g,g' continuous}$$ Has a maximum value for $\min{(g')}$ of $\max{(g)} - \min{(g)}$. This when $\min{(g)}=g(0)$, $\max{(g)}=g(1)$ and $g$ is a straight line. If I haven't understood, the question seems trivial. What have I got wrong? Last edited by v8archie; June 27th, 2017 at 07:02 AM.
 June 27th, 2017, 07:19 AM #7 Member   Joined: May 2017 From: France Posts: 57 Thanks: 1 Are you say : $\displaystyle \max(g)-\min(g) \geq \min(g')$ ? What about $\displaystyle g(x)=\sinh(x)$ ?
June 27th, 2017, 08:33 AM   #8
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Quote:
 Originally Posted by Dattier Are you say : $\displaystyle \max(g)-\min(g) \geq \min(g')$ ?
I believe so.

$\sinh 1 - \sinh 0 \approx 1.18 - 0 = 1.18$
$\min{(\cosh x)}=1$

Last edited by v8archie; June 27th, 2017 at 08:36 AM.

 June 27th, 2017, 09:05 AM #9 Member   Joined: May 2017 From: France Posts: 57 Thanks: 1 edit : I must think What about this inequality $\displaystyle 4(\max(g)-\min(g))\geq \min(g'')$, when $\displaystyle g \in C^2([0,1])$ Last edited by Dattier; June 27th, 2017 at 09:16 AM.
 June 27th, 2017, 09:27 AM #10 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,327 Thanks: 2451 Math Focus: Mainly analysis and algebra That sounds plausible. Again consider the case where $g''$ is constant (giving the maximum value for $\min{(g'')}$.

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