There is no way you can add a whole bunch of positive terms together and get zero.
Hence, if 1 + x0/1! + x02/2! + .... + x0n-1/(n-1)! = 0,
then x0<0.
En''(x)= 1 + x/1! + x2/2! + .... + xn-2/(n-2)! =En'(x)- xn-1/(n-1)!
So, E''(x0) = En'(x0)- x0n-1/(n-1)!
= - x0n-1/(n-1)!, which is...