Second Derivative at Maximum is Negative/Mistake

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

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:

\(\ds \map {f'} x\)

\(=\)

\(\ds 4 x^3\)

\(\ds \leadsto \ \ \)

\(\ds \map {f} x\)

\(=\)

\(\ds 12 x^2\)

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.

Sources

• 1963: Charles Fox: An Introduction to the Calculus of Variations (2nd ed.) ... (previous) ... (next): Chapter $\text I$. The First Variation: $1.2$. Ordinary maximum and minimum theory