f′>0 ⇒ increase, f′<0 ⇒ decrease
If f is continuous on [a;b] and differentiable on (a;b), and f′(x)>0 for all x∈(a;b), then f strictly increases on [a;b] (similarly for f′<0 and strict decrease).
Stationary points: f′(x0)=0 — “candidates” for extrema (often interior, if f′ changes sign nearby).
Fermat’s theorem (necessary condition): if x0 is a local extremum inside an interval of differentiability, then f′(x0)=0 or the derivative is not defined there.
f′(x0)=0 does not guarantee a local extremum (f(x)=x3 at 0: f′(0)=0, but there is no extremum — an inflection there)
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