In any case, we have the trivial solution x=0, and the two you've found in e(i).
Here are a few questions you might like to consider:
- What happens when one (or more!) of the two solutions you found coincides with the trivial solution? When does this occur? (Note that you have already done this for x = 0). Is it possible to limit these instances to a finite set, then generalise their behaviour around x = 0?
- Check the behaviour of the second derivative around x = 0 (both the left-hand and right-hand limits)
- As e
-x gets large, depending on the parity of n, we approach a function similar to x
n. Does the parity also have an impact on how many inflection points the function has?
Things that might tip you off to checking the above include the finite number of cases, the clear distinction between cases etc. It's also a lot easier to start at small values of n, like 2 or 3 and then generalise for the sets mentioned above that include them.
As for analytical vs. several graphs - I'd like to think this way of thinking is analytical enough (you only really need to draw 2-3 graphs, and only if you really do need them). Since n is just an integer, an exhaustive graphing method is not going to be good enough as you need to generalise for some subset of the integers.
Hope this makes sense - feel free to drop any other queries or followups