May 9th, 2017, 08:18 PM  #1 
Senior Member Joined: Apr 2008 Posts: 177 Thanks: 3  how to find inflection points
I have a lot of trouble with this problem. (ex) Consider the function g(x) = x*e^(x) + b*e^(x) where b is a constant. Find all values of b if any for which the graph of g has a point of inflection in 0 < x < infinity. my attempt The second derivative of g is g''(x) =  e^(x) (x  (2  b)) Setting g''(x) = 0 and solving for x give x= 2  b. Now, I have a problem. I cannot find b because there are two unknowns, which are x and b. Can someone explain how to do it? Thanks. 
May 9th, 2017, 09:14 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,653 Thanks: 840 
$x_{inf} = 2b$ $x > 0$ $x_{inf}>0 \Rightarrow 2b > 0$ $b < 2$ 
May 10th, 2017, 04:41 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,875 Thanks: 766 
The problem did not ask you to find a value of b, it asks you to find all values of b for which x is greater than 0. In other words, solve x= 2 b> 0.

May 11th, 2017, 11:31 AM  #4 
Senior Member Joined: Apr 2008 Posts: 177 Thanks: 3 
Thank you, romsek and Country Boy.


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