f″(x) = concavity of f(x)

Meaning: $f\prime \prime \left(x\right)$

The 2 ^nd Derivative tells you about the concavity of the original function, $f\left(x\right)$.

$f\prime \prime \left(x\right)=\mathit{positive }(+)=\mathit{concave\; up }\left(\mathit{like\; a\; cup}\right)$

$f\prime \prime \left(x\right)=\mathit{negative }(–)=\mathit{concave\; down }\left(\mathit{like\; a\; frown}\right)$

$f\prime \prime \left(x\right)=\mathit{zero }\left(0\right)=\mathit{not\; concave\; up\; or\; concave\; down}=\mathit{possible }\mathit{inflection\; point}$

Note: Having the 2 ^nd Derivative equal zero is only one part of the requirement to be an inflection point. The original function, $f\left(x\right)$, must also change concavity at that point. From concave up to concave down or from concave down to concave up.