I understand these concepts (increasing, decreasing, maximum, minimum, etc.) at a very high level, but it is useful to define them precisely.

High level of $f_{â€˛}(x)$ and $f_{â€˛â€˛}(x)$

Intuitively, $f_{â€˛}(x)$ represents the rate of change of the values of $f(x)$ (or think of it as the slope of the tangent lines to $f(x)$. We use $f_{â€˛}(x)$ to understand when a function is increasing and decreasing, as well as find extrema values.

$f_{â€˛â€˛}(x)$ is a bit harder to grasp. It describes the rate of change of the tangent line to $f(x)$.
Think of it related to Newtonâ€™s law,
$s_{â€˛â€˛}(t)=v_{â€˛}(t)=a(t)$
The second derivative provides information about the curve behavior, which we call concavity.

Increasing and Decreasing, Monotonic

A function $f$ is increasing if $f(x_{2})>f(x_{1})$ whenever $x_{2}>x_{1}$.
A function $f$ is decreasing if $f(x_{2})<f(x_{1})$ whenever $x_{2}>x_{1}$.
We can define these in terms of the derivative,

Note: The converse is NOT true. If $f$ is increasing we may still find that $f_{â€˛}(x)=0$ at some isolated points.#idontunderstand

Maxima and Minima (Extrema), Critical Point

A function $f$ has an absolute maximum at $c$ if $f(c)â‰Ąf(x)$ for all $x$ in the domain of $f$.
A function $f$ has an absolute minimum at $c$ if $f(c)â‰¤f(x)$ for all $x$ in the domain of $f$.

Algorithm for identifying Global Extrema: Closed Interval Method

We use the idea that if $c$ is a global maximum or minimum of $f$, one of three things must be true:

$f_{â€˛}(c)=0$ (a critical point)

$f_{â€˛}(c)$ is undefined (a critical point)

$c$ is an endpoint of the interval.

Thus, we develop an algorithm for this.

Algorithm for identifying Local Extrema: First Derivative Test

The second derivative test also works.

Concavity, Inflection Point

If the concavity of $f(x)$ changes at a point $(x_{0},f(x_{0}))$, then we call this point an inflection point.