August 4th, 2019, 12:28 PM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 708 Thanks: 96  Minimal value
Find the minimal value of $\displaystyle f(x)=e^{x} +x\cdot e^{x} .$
Last edited by idontknow; August 4th, 2019 at 12:39 PM. 
August 4th, 2019, 01:12 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,834 Thanks: 733 
$f(x)\gt 0$ for all $x$. $f(x)\to 0$ as $x\to \infty$. That is the minimum.

August 4th, 2019, 07:14 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 21,015 Thanks: 2250 
Isn't 0 a limiting value (and greatest lower bound) rather than a minimum value?

August 5th, 2019, 12:34 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,834 Thanks: 733 
It depends on a precise definition of minimum. Greatest lower bound is precise, minimum is not.

August 5th, 2019, 11:57 PM  #5 
Senior Member Joined: Dec 2015 From: somewhere Posts: 708 Thanks: 96  

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