1. Curve Sketching

I'm having trouble finding the point of inflection of the graph

f(x)=(8e^x)/(e^(2x)+4)

so I cannot find the points of inflection even though I know the procedure:

f''(x)=(8e^x)/(e^(2x)+4)

f''(x)= 0 and f''(x) = undefined both do not exist...soo there is no point of inflection right? no.. there has to be BECAUSE, when i graph the function, concavity is changing.
from (-infinity, ln2) we have concave up and from (infinity,ln2) concave down. now... how do i find the point of inflection if i know that concavity is changing, but the second derivative does not allow me to find the point?

other info:
max point (ln2,2)
HA=y=0
yint=1.909

2. $\displaystyle f''(x)=\frac{8e^x(e^{4x}-24e^{2x}+16)}{(e^{2x}+4)^3}$

$\displaystyle f''(x)=0\Rightarrow e^{4x}-24e^{2x}+16=0$

Put $\displaystyle e^{2x}=t$ and solve the quadratic.

3. i end up getting t= 23 and t=0.7

when i graph the function, t=0.7 does look like a PI but t=23 seems way off.. what did you get?

4. can anyone verify this???