91Ó°ÊÓ

Extreme points or extreme values of a function are sometimes known as the function's minimum and maximum. They can be regional or global in scope.

The maximum that operates on functions needs to be determined.

Prove that the maximum operator which operates on the functions of x is a linear operator or not from the following definition.

An operator O represents a linear operator if the following relations are satisfied.

$$\begin{gathered} \mathrm{O}(\mathrm{A}+\mathrm{B})=\mathrm{O}\left(\mathrm{A}\right)+\mathrm{O}\left(\mathrm{B}\right)\% \\ \mathrm{O}\left(\mathrm{kA}\right)=\mathrm{kO}\left(\mathrm{A}\right)\% \end{gathered}$$

where k is a number, A and B are numbers, functions, vectors, etc.

Conclude that the maximum operator which operates on the functions of x which is given by max ( f ( x ) ) . Use the definition of the maximum function.

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)≤\max \left(\mathrm{f}\right(\mathrm{x}\left)\right)\% \\ \mathrm{g}\left(\mathrm{x}\right)≤\max \left(\mathrm{g}\right(\mathrm{x}\left)\right) \end{gathered}$$

Take the sum of the previous equations.

$\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)≤\max \mathrm{f}\left(\mathrm{x}\right)+\max \mathrm{g}\left(\mathrm{x}\right)$

The following triangle inequality is deduced.

$\max \left\{\mathrm{f}\right(\mathrm{x})+\mathrm{g}(\mathrm{x}\left)\right\}≤\max \mathrm{f}\left(\mathrm{x}\right)+\max \mathrm{g}\left(\mathrm{x}\right)$

It has been shown that the maximum operator with the objects being operated on the functions of x does not represent a linear operator. For example, let f(x), t(x) and g(x) be two functions to be defined on a domain D. The maximum functions are defined as follows

$\max (\mathrm{f},\mathrm{t})\left(\mathrm{x}\right)=\left\{\begin{array}{l}\mathrm{f}\left(\mathrm{x}\right),\mathrm{f}\left(\mathrm{x}\right)â‰\yen \mathrm{t}\left(\mathrm{x}\right)\\ \mathrm{t}\left(\mathrm{x}\right),\mathrm{f}\left(\mathrm{x}\right)â‰\yen \mathrm{t}\left(\mathrm{x}\right)\end{array}\right.\mathrm{and},\max (\mathrm{f},\mathrm{g})\left(\mathrm{x}\right)=\left\{\begin{array}{l}\mathrm{f}\left(\mathrm{x}\right),\mathrm{f}\left(\mathrm{x}\right)â‰\yen \mathrm{g}\left(\mathrm{x}\right)\\ \mathrm{g}\left(\mathrm{x}\right),\mathrm{f}\left(\mathrm{x}\right)â‰\yen \mathrm{g}\left(\mathrm{x}\right)\end{array}\right.$

Choose$\mathrm{t}\left(\mathrm{x}\right)â‰\yen \mathrm{f}\left(\mathrm{x}\right)\mathrm{and}\mathrm{t}\left(\mathrm{x}\right)â‰\yen \mathrm{g}\left(\mathrm{x}\right)$

$$\begin{gathered} \max (\mathrm{f},\mathrm{t})\left(\mathrm{x}\right)=\mathrm{t}\left(\mathrm{x}\right),\mathrm{where}\mathrm{t}\left(\mathrm{x}\right)â‰\yen \mathrm{f}\left(\mathrm{x}\right) \\ \max (\mathrm{f},\mathrm{g})\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{x}\right),\mathrm{where}\mathrm{t}\left(\mathrm{x}\right)â‰\yen \mathrm{g}\left(\mathrm{x}\right) \end{gathered}$$

Deduce the possible relationship.

$$\begin{gathered} \max (\mathrm{f}+\mathrm{g},\mathrm{t})\left(\mathrm{x}\right)=(\mathrm{f}+\mathrm{g})\left(\mathrm{x}\right) \\ =\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right) \\ ≠\mathrm{t}\left(\mathrm{x}\right)+\mathrm{t}\left(\mathrm{x}\right) \end{gathered}$$

Therefore, the maximum operator which operates on functions of x does not represent a linear operator.