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It is said that the logarithm is a monotonically increasing function, hence the logarithm of a function achieves its maximum value at the same points as the function itself.

Is there a similar property for monotonically decreasing functions and minimization, e.g, the minimum points for $f(x)$ and $e^{-f(x)}$ are the same?

Adam I.
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1 Answers1

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Let $f$ be some function

  • If $g$ is strictly monotonically increasing, then $g(f(x))$ has the same minimizer and maximizer as $f(x)$

  • If $g$ is strictly monotonically decreasing, then the minimizers of $g(f(x))$ are the maximizer of $f(x)$ and the maximizers of $g(f(x))$ are the minimizer of $f(x)$.

Surb
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