Let $f\,$ be suitably differentiable near to $a\,$. Then $f\,$ has a local maximum or minimum at $a\,$ only if $f^{\prime}(a)=0\,$. Furthermore,

(a) if $f^{\prime}(a)=0\,$ and $f^{\prime\prime}(a)>0\,$, then $f\,$ has a local minimum at $a\,$;

(b) if $f^{\prime}(a)=0\,$ and $f^{\prime\prime}(a)<0\,$, then $f\,$ has local maximum at $a\,$; whereas

(c) if $f^{\prime}(a)=f^{\prime\prime}(a)=0$ and $f^{\prime\prime\prime}(a)\neq 0$, then $f$ has an inflexion at $a$.