need help on finding inflection points [ApCalc]?

Question

the equations is y=xe^x. i'm not sure if my 2nd derivative is right..& if it is....when you set it = to 0 ..lookssss extremely confusing. i'm not even sure if my f'(x) is right

f'(x)= e^x+xe^x
my f''(x) = e^x(xe^x)+e^x(e^x+xe^x)
so u're supposed to do?
e^x(xe^x)+e^x(e^x+xe^x)=0 ??
idk if anyone could help it would be greatly appreciated. this is the 1st problem of my hw lol ..i might have to post more if i don't get the other ones

y=x sqrt(9-x^2)
y=x^3(4-x)
y=x^1/3(x-4)
y=x^1/2(x+3)

i absolutely have no idea how to do these -_- if anyone could help me step by step?? >.<

Answer 1

y' = e^x(x+1)
y'' = e^x(x+2) =0
x = -2
f(-2) = -2/e^2
(-2, -2/e^2) is the point of inflection.

In a similar way, you can do the other problems by solving y'' = 0, and check if there is a sign change from both sides.

Answer 2

f '(x+ = e^x +xe^x
f "(x) = 2e^x + xe^x = e^x(2+x)
Now it is clear that f "(x) is positive x >-2 and is negative when x<-2. f "(x) is 0 if x = -2. It follows that the curve is concave upward when when x >-2 and concave downward wheb x < - 2. Thus x= - 2 is a point of inflection.