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May 9th, 2017, 07:18 PM   #1
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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.
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May 9th, 2017, 08:14 PM   #2
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$x_{inf} = 2-b$

$x > 0$

$x_{inf}>0 \Rightarrow 2-b > 0$

$b < 2$
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May 10th, 2017, 03:41 AM   #3
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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.
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May 11th, 2017, 10:31 AM   #4
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Thank you, romsek and Country Boy.
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