Point with Zero Second Derivative is not necessarily Point of Inflection

Theorem

Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$.

Let

$\map {f' '} \xi = 0$

where $\map {f' '} \xi$ denotes the second derivative of $f$ at $\xi \in \openint a b$.

Then it is not necessarily the case that $f$ has a point of inflection at $\xi$.

Proof

Consider the function:

$f: \R \to \R: \forall x \in \R: \map f x = x^4$

This has a local minimum at $x = 0$ at which $\map {f' '} x = 0$.

But $x = 0$ is not a point of inflection of $f$.

$\blacksquare$

Also see

• Second Derivative at Point of Inflection

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inflection
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): turning point
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inflection
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): turning point