Single-variable optimization, definitions

Summary

Minimum and maximum points

• $f:A→B$ and $x_0∈A$
• $x_0$ 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 $x∈A,x≠x_0$ .
• Examples:
• 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

• $f:A→B$ and $x_0∈A$
• $x_0$ 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

• $f:A→B$ and $x_0∈A$
• 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 $f\left( x \right)=x^2$ .