Perhaps this is a silly question, but I haven't been able to find a clear answer anywhere as to what exactly the steps are for using derivatives to find the shape of a graph (I'm having difficulty even expressing what I mean properly). I've done my best to take information from my textbook and on the internet and create a list of what I think is supposed to be going on.
So here it goes:
- used to find $f'(x)$
- produces the y-value for a critical number found using $f'(x)$
- produces the y-value for the inflection point found using $f''(x)$
- finds $f''(x)$
- finds critical numbers
- shows what intervals a function is increasing or decreasing on by using the roots of $f'(x)$
- First derivative test: finds a local maximum or minimum
- finds where $f(x)$ is a maximum or minimum by evaluating $f''(x)$ by the roots of $f'(x)$ (which takes place of the first derivative test)
- by using the critical numbers of $f'(x)$, $f''(x)$ can find where the graph is concave up or concave down.
- gives where the inflection points are by finding where $f''(x)=0$, and then evaluating $f(x)$ at these numbers to find the point.
Any help I can get on this will be greatly appreciated!