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Let us consider the following function $f:X\rightarrow{R}$. Let us denote by $f_{max}$ and $f_{min}$, respectively, maximum and minimum values of $f$ function. Let us consider the following monotonic function $g:X\rightarrow{R}$. By using monotonic transformation of $f$ we obtain the following function $h:X\rightarrow{R}$, where $h=g(f)$. Let us denote by $h_{max}$ and $h_{min}$, respectively, maximum and minimum values of $h$ function.

What is the relationship between $f_{max}$ and $h_{max}$ and $f_{min}$ and $h_{min}$?

P.S. I know that for monotonic transformation the extreme values match. In what cases it is true?

David
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  • Do you mean $g\colon{\Bbb R}\to {\Bbb R}$? It is easy to see that the optimal solutions, if exists, $x_{opt}$ are the same for $f$ and $h$. – A.Γ. May 14 '18 at 11:07
  • The question can be transformed in the following way: what is the relationship between $f(x_{max})$ and $h(x_{max})$? That is, what is the relationship between $f$ and $h$ functions' extreme values? – David May 14 '18 at 11:49

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