Second Derivative at Maximum is Negative/Mistake

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Source Work

1963: Charles Fox: An Introduction to the Calculus of Variations (2nd ed.):

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


... 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.


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:

\(\ds \map {f'} x\) \(=\) \(\ds 4 x^3\)
\(\ds \leadsto \ \ \) \(\ds \map {f''} x\) \(=\) \(\ds 12 x^2\)

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.