Definition:Turning Point

Definition

Local Maximum

Let $f$ be a real function defined on an open interval $\openint a b$.

Let $\xi \in \openint a b$.

Then $f$ has a local maximum at $\xi$ if and only if:

$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \le \map f \xi$

That is, if and only if there is some subinterval on which $f$ attains a maximum within that interval.

Local Minimum

Let $f$ be a real function defined on an open interval $\openint a b$.

Let $\xi \in \openint a b$.

Then $f$ has a local minimum at $\xi$ if and only if:

$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \ge \map f \xi$

That is, if and only if there is some subinterval on which $f$ attains a minimum within that interval.

Also known as

A turning point is also known as a bend point.

Also see

• Results about turning points can be found here.

Sources

• 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): turning point