# 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

## 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