Sign of f′ between the critical points
f′ > 0: f is climbing. f′ < 0: f is falling. A critical point is a max if f′ goes + then −, a min if − then +, and nothing (a saddle) if the sign does not change.
Critical points and verdicts
Critical point = where f′(x) = 0 (or f′ undefined). Candidates only. Nothing is a max or min until the sign of f′ around it is checked.
Classify by the sign change of f′, or by f″: f″ < 0 max, f″ > 0 min, f″ = 0 tells you nothing.
A squared factor in f′, like (x + 1)², never changes sign, so that critical point is a saddle, not an extremum.
Global on a closed interval = compare f at every critical point inside the interval and at both endpoints. The endpoints often win.