f′′f'' as “acceleration of slope”
If f′′>0f''>0 on an interval, the graph is concave up (a cup “holds water” in the usual axis orientation). If f′′<0f''<0 — concave down. Inflection points are suspected where f′′f'' changes sign or ceases to exist while ff remains continuous. Graph-study order (draft): D(f)D(f), continuity, asymptotes → signs of f′,f′′f',f'' → extrema, concavity, inflections → symmetries if any.
f′′(x0)=0f''(x_0)=0 does not mean inflection by itself — you need a sign change of f′′f''
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