f′′ as “acceleration of slope”
If f′′>0 on an interval, the graph is concave up (a cup “holds water” in the usual axis orientation). If f′′<0 — concave down.
Inflection points are suspected where f′′ changes sign or ceases to exist while f remains continuous.
Graph-study order (draft): D(f), continuity, asymptotes → signs of f′,f′′ → extrema, concavity, inflections → symmetries if any.
f′′(x0)=0 does not mean inflection by itself — you need a sign change of f′′
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