The important points:
- This function is odd.
- The slope at 0 is infinite!
- There is an inflection point at 0 (the only one), when the second derivative is undefined.
- Regarding the fact that the second derivative is never 0 (first line of the solution, far right): I prefer that one say "there are no real solutions" rather than "false" (which is what the calculator reports!). Sometimes the calculator says "false" when there are, in fact, complex solutions (e.g., if you try to solve x2+1=0 for example).
- This is a labor-intensive problem, but it is basically solving four equations in four unknowns.
- By hand this is tedious: there's a whole branch of mathematics devoted to this problem (linear algebra). It's very important in many branches of science, and computer science as well.