4.17 Proof of the sign test for stationary points

(Not examinable.)

It is clear that $f\,$ cannot have a local maximum at $a\,$ if the tangent of $f$ is sloping upwards or downwards. Similarly $f$ cannot have a local minimum, so local maxima and minima can only occur at stationary points. (This can also be proved using Taylor’s expansion.)

Suppose now that we have a stationary point, so $f^{\prime}(a)=0\,$. Then the Taylor expansion about $a\,$, reduces to

In cases (a) and (b), we have $f^{\prime\prime}(a)\neq 0\,$; so the series reduces to an approximation, for small $|h|\,$,

since the remaining terms in the series are small in comparison to the term involving $f^{\prime\prime}(a)\,$. We have the following cases.