You can almost tell that the #e^18# is a distractor. It doesn't matter what the exponent is because any #e^C# is just a constant.
The first derivative gives you the slope. The second gives you the concavity.
Since that is the case, when the second derivative at some #x# value is positive, it's concave up and vice versa. A function with an inflection point is either concave up, concave down, or neither. Therefore, if there's somewhere that the second derivative is equal to #0#, then it must be neither concave up nor down, and so it must be an inflection point.
#d/(dx)[f(x)] = d/(dx)[e^18 x^3] = e^18 (3x^2)#
Simply from writing out #f(x)#, you can tell that you have a basic cubic function with no vertical or horizontal shifts. So, I would expect only one inflection point, and it's probably at #x = 0#.
#d/(dx)[e^18 (3x^2)] = e^18 (6x)#
Now let's just set this to #0#:
#e^18 (6x) = 0#
#color(blue)(x = 0)#
...Yep, that's really the only one.
