Maxima, minima and critical points
A function has infinitely many inputs. How can its derivative point to the few places where a peak or a valley can sit, and tell which is which?
Where you are 4 inside
What this area is about
Algebra found highest and lowest points by reading a graph or completing the square. Calculus finds them from the derivative, with proof that none were missed. A smooth peak or valley inside the domain has a level tangent, so the search for extreme values shrinks from infinitely many inputs to a handful.
How these fit together
Definitions and the key fact. Critical points and local versus absolute extrema defines the inputs c where f'(c) = 0 or f'(c) does not exist, and proves that every local extremum sits at one.
Results. Increasing and decreasing from the derivative turns the sign of f' on an interval into rising or falling, using a sign chart cut at the critical points. The first and second derivative tests then sort each critical point into peak, valley or neither by whether f' changes sign there; at a level tangent with f''(c) \ne 0, the sign of f''(c) settles the same question.
Procedure. Extrema on a closed interval finds the absolute maximum and minimum of a function continuous on [a, b] by comparing f at the critical points and at both endpoints.
A path through it
Critical points first. Then sign charts, then the derivative tests. The closed interval method needs only critical points, so it can come right after them or last.
Finish here and you can do optimization, curve sketching from sign charts and Rolle’s theorem.