Point of local extremum

Definition

A point of local extremum refers to a point in the interior of the domain of a function that is either a point of local maximum or a point of local minimum. Both these are defined below.

Point of local maximum

A point $c$ in the interior of the domain of a function $f$ is a point of local maximum if there exists a value $\delta >0$ such that $f(x)\le f(c)$ for all $x\in (c-\delta ,c+\delta )$ (i.e., all $x$ satisfying $|x-c|<\delta$).

The value $f(c)$ is termed a local maximum value.

Point of local minimum

A point $c$ in the interior of the domain of a function $f$ is a point of local maximum if there exists a value $\delta >0$ such that $f(x)\ge f(c)$ for all $x\in (c-\delta ,c+\delta )$ (i.e., all $x$ satisfying $|x-c|<\delta$).

The value $f(c)$ is termed a local minimum value.

Variations

Variation name	Definition
Point of strict local maximum	A point $c$ in the interior of the domain of a function $f$ is a point of strict local maximum if there exists a value $\delta >0$ such that $f(x)<f(c)$ for all $x\in (c-\delta ,c+\delta )\setminus \{c\}$ (i.e., all $x$ satisfying $0<|x-c|<\delta$).
Point of strict local minimum	A point $c$ in the interior of the domain of a function $f$ is a point of strict local minimum if there exists a value $\delta >0$ such that $f(x)>f(c)$ for all $x\in (c-\delta ,c+\delta )\setminus \{c\}$ (i.e., all $x$ satisfying $0<|x-c|<\delta$).
Point of local maximum from the left	A point $c$ in the domain of a function $f$ is a point of local maximum from the left if there exists a value $\delta >0$ such that $f(x)\le f(c)$ for all $x\in (c-\delta ,c)$ (i.e., all $x$ satisfying $0<c-x<\delta$).
Point of local maximum from the right	A point $c$ in the domain of a function $f$ is a point of local maximum from the right if there exists a value $\delta >0$ such that $f(x)\le f(c)$ for all $x\in (c,c+\delta )$ (i.e., all $x$ satisfying $0<x-c<\delta$).
Point of local minimum from the left	A point $c$ in the domain of a function $f$ is a point of local minimum from the left if there exists a value $\delta >0$ such that $f(x)\ge f(c)$ for all $x\in (c-\delta ,c)$ (i.e., all $x$ satisfying $0<c-x<\delta$).
Point of local minimum from the right	A point $c$ in the domain of a function $f$ is a point of local minimum from the right if there exists a value $\delta >0$ such that $f(x)\ge f(c)$ for all $x\in (c,c+\delta )$ (i.e., all $x$ satisfying $0<x-c<\delta$).
Point of strict local maximum from the left	A point $c$ in the domain of a function $f$ is a point of strict local maximum from the left if there exists a value $\delta >0$ such that $f(x)<f(c)$ for all $x\in (c-\delta ,c)$ (i.e., all $x$ satisfying $0<c-x<\delta$).
Point of strict local maximum from the right	A point $c$ in the domain of a function $f$ is a point of strict local maximum from the right if there exists a value $\delta >0$ such that $f(x)<f(c)$ for all $x\in (c,c+\delta )$ (i.e., all $x$ satisfying $0<x-c<\delta$).
Point of strict local minimum from the left	A point $c$ in the domain of a function $f$ is a point of strict local minimum from the left if there exists a value $\delta >0$ such that $f(x)>f(c)$ for all $x\in (c-\delta ,c)$ (i.e., all $x$ satisfying $0<c-x<\delta$).
Point of strict local minimum from the right	A point $c$ in the domain of a function $f$ is a point of strict local minimum from the right if there exists a value $\delta >0$ such that $f(x)>f(c)$ for all $x\in (c,c+\delta )$ (i.e., all $x$ satisfying $0<x-c<\delta$).

Facts

• Point of local extremum implies critical point
• Critical point not implies point of local extremum
• Local maximum from the left implies left hand derivative is nonnegative if it exists
• Local maximum from the right implies right hand derivative is nonpositive if it exists
• Local minimum from the left implies left hand derivative is nonpositive if it exists
• Local minimum from the right implies right hand derivative is nonnegative if it exists