In derivative,
If $f'(x)$ is rising at $f'(x)$ = 0, there's a local minima in $f(x)$.
If $f'(x)$ is falling at $f'(x)$ = 0, there's a local maxima in $f(x)$.
If $f''(x)$ is rising at $f''(x)$ = 0, there's a local minima in $f'(x)$ and $f(x)$ is falling.
If $f''(x)$ is falling at $f''(x)$ = 0, there's a local maxima in $f'(x)$ and $f(x)$ is rising.
We can graph the function out to see whether it's falling or rising, A simple check at the position very close the root may suffice to see whether we got a positive or negative number, and conclude whether a function is falling or rising. Derivative can be used to find roots, maxima, minima, rising slope, and falling slope.
In numerical method, (or more precisely, for a computer program) we can use Bisection method, Newton-Raphson method to approximate roots of a function.
Now, what about other features such as local maxima, minima and whether the function is rising or falling? I am looking for an algorithm for approximating these critical points. What are some of the numerical method names I should be looking for?