Single-variable optimization, definitions

Summary

Minimum and maximum points

\(x_0\) is called a maximum point for \(f\) if \(f(x)≤f(x_0)\) for all \(x∈A\) .
\(x_0\) is called a strict maximum point for \(f\) if \(f(x)<f(x_0)\) for all \(x∈A,x≠x_0\) .
\(x_0\) is called a minimum point for \(f\) if \(f(x)≥f(x_0)\) for all \(x∈A\) .
\(x_0\) is called a strict minimum point for \(f\) if \(f(x)>f(x_0)\) for all \(x∈A,x≠x_0\) .

If \(f(x)=x^2\) then \(x_{min}=0\) is a strict minimum point of \(f\)
If \(f(x)=-x^2\) then \(x_{max}=0\) is a strict maximum point of \(f\)
If \(f(x)=1\) then every point is maximum point and every point is a minimum point of \(f\) but none of them are strict.
If \(f(x)=x\) then \(f\) has no maximum points and no minimum points
If \(f(x)=x\) with domain \((0,1)\) then \(f\) has no maximum points and no minimum points
If \(f(x)=x\) with domain \([0,1]\) then \(x_{min}=0\) is a strict minimum point of \(f\) and \(x_{max}=1\) is a strict maximum point of \(f\) .

A point that is either a maximum point or a minimum point is called a extreme point or an optimal point of \(f\) .

Minimum and maximum values

If \(x_0\) is a maximum point of \(f\) then \(f(x_0)\) is called the maximum value of \(f\) .
If \(x_0\) is a minimum point of \(f\) then \(f(x_0)\) is called the minimum value of \(f\) .
Examples:

If \(f(x)=x^2\) then the minimum value of \(f\) is \(0\) .
If \(f(x)=-x^2\) then the maximum value of \(f\) is \(0\) .
If \(f(x)=1\) then the minimum value of \(f\) and the maximum value of \(f\) is \(1\) .
If \(f(x)=x\) then \(f\) has no minimum or maximum values.
If \(f(x)=x\) with domain \((0,1)\) then \(f\) has has no minimum or maximum values.
If \(f(x)=x\) with domain \([0,1]\) then the minimum value of \(f\) is \(0\) and the maximum value of \(f\) is \(1\) .

Local minimum and maximum points

\(x_0\) is called a local maximum point for \(f\) if there exits an open interval \((a,b)\) containing \(x_0\) such that \(f(x)≤f(x_0)\) for all \(x∈(a,b)∩A\) .
\(x_0\) is called a strict local maximum point for \(f\) if there exits an open interval \((a,b)\) containing \(x_0\) such that \(f(x)<f(x_0)\) for all \(x∈(a,b)∩A,x≠x_0\) .
\(x_0\) is called a local minimum point for \(f\) if there exits an open interval \((a,b)\) containing \(x_0\) such that \(f(x)≥f(x_0)\) for all \(x∈(a,b)∩A\) .
\(x_0\) is called a strict local minimum point for \(f\) if there exits an open interval \((a,b)\) containing \(x_0\) such that \(f(x)>f(x_0)\) for all \(x∈(a,b)∩,x≠x_0\) .

Every maximum (minimum) point is a local maximum (minimum) point but the opposite is not true.
A point that is either a local maximum point or a local minimum point is called a local extreme point or a local optimal point of \(f\) .

Local minimum and maximum values

If \(x_0\) is a local maximum point of \(f\) then \(f(x_0)\) is called a local maximum value of \(f\) .
If \(x_0\) is a local minimum point of \(f\) then \(f(x_0)\) is called a local minimum value of \(f\) .

Stationary points

If \(f'(x_0)=0\) then \(x_0\) is called a stationary or critical point for \(f\)
Examples:

If \(f(x)=x^2\) or \(f(x)=-x^2\) then \(x_0=0\) is a critical point for \(f\) .
if \(f(x)=1\) then every point is a critical point for \(f\) .
If \(f(x)=x\) then \(f\) has no critical points, no matter the domain.
If \(f(x)=|x|\) then \(x_{min}=0\) is a strict minimum point for \(f\) but it is not a critical point.

Notation. Some authors define minimum point, maximum point and stationary point as a point on the graph of \(f\) , including both the \(x\) - and the \(y\) -ccordinate. In this notation, \(\left( 0,0 \right)\) is a stationary point of \(f\left( x \right)=x^2\) .