4.18 Cases of the sign test

In cases (a) and (b) $f(a)+f^{\prime\prime}(a)h^{2}/2$ gives a parabola in the variable $h$; pointing upwards in (a), downwards in (b).

(a) If $f^{\prime\prime}(a)>0\,$, then $f(a+h)>f(a)\,$ for all $h\neq 0$ such that $|h|\,$ is small and $f\,$ has a local minimum; whereas

(b) if $f^{\prime\prime}(a)<0\,$, then $f(a+h)<f(a)\,$ for all $h\neq 0$ such that $|h|\,$ is small and $f\,$ has a local maximum.

(c) In this case $f^{\prime}(a)=f^{\prime\prime}(a)=0$, so the Taylor expansion becomes

with $f^{\prime\prime\prime}(a)\neq 0$. So $f(a+h)-f(a)$ can be made positive or negative; hence we have an inflexion.