Find Intervals, where Function is Convex or Concave and Inflection Points

[knowLittle]

Homework Statement

y= (x^2 -7) e^x

The Attempt at a Solution

I'm trying to find inflection points by setting the second derivative=0
I found that the derivative is:
$2xe^{x}+x^{2}e^{x}-7e^{x}=0$
$e^{x}[2x+x^{2}-7]=0$
Then, the 2nd derivative:
$e^{x}[(x-1)(x+5)]=0$, then the inflection points are at x=- infinity; 1; -5.
Where $e^{x}=0$, happens when x=-infinity
Is it correct to use - infinity as a value of x ?

 

[Ray Vickson]

Homework Statement

y= (x^2 -7) e^x

The Attempt at a Solution

I'm trying to find inflection points by setting the second derivative=0
I found that the derivative is:
$2xe^{x}+x^{2}e^{x}-7e^{x}=0$
$e^{x}[2x+x^{2}-7]=0$
Then, the 2nd derivative:
$e^{x}[(x-1)(x+5)]=0$, then the inflection points are at x=- infinity; 1; -5.
Where $e^{x}=0$, happens when x=-infinity
Is it correct to use - infinity as a value of x ?

No, it's not. Basically, an inflection point x0 is a point where the behavior of f(x) changes from convex to concave (or opposite) as x increases through x0. How do you increase from values < -∞ to > -∞ (that is, how do you pass from one side of -∞ to the other)?

RGV