Second Derivative at Maximum is Negative/Mistake

Source Work

 * Chapter $\text I$. The First Variation:
 * $1.2$. Ordinary maximum and minimum theory
 * $1.2$. Ordinary maximum and minimum theory

Mistake

 * ... it follows that at a maximum $\map {f} a$ is negative and ... that at a minimum $\map {f} a$ is positive. Alternatively at a maximum $\map {f'} x$ is a decreasing function of $x$ and at a minimum $\map {f'} x$ is an increasing function of $x$. Thus it is possible to discriminate quite easily between maxima and minima.

Correction
Not necessarily.

Let $\map f x = x^4$.

Then we have that $f$ has a local minimum at $x = 0$.

However, by Power Rule for Derivatives:

both of which are definitely zero at $x = 0$.

Hence, while the above statement is true in general, it needs to be pointed out that there are special cases which need to be investigated carefully.