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  • $\begingroup$ David, I think that these links might be of interest to you since they are the demonstrations of the mean value theorem and intermediate value theorem. Not sure if you knew about these or not. Also, a clarifying question: are you trying to build a routine from scratch that does the trick or do you want to use built-in functions? $\endgroup$ Commented Jan 3, 2023 at 6:52
  • $\begingroup$ I'm interested in using existential quantifiers (ForAll, Exists, ...) and such to "prove" this relation in Mathematica. In another context: Reduce[\!(* SubscriptBox["[ForAll]", RowBox[{"s", ",", RowBox[{"s", "[Element]", TemplateBox[{}, "Reals"]}]}]](* SubscriptBox["[Exists]", RowBox[{"t", ",", RowBox[{"t", "[Element]", TemplateBox[{}, "Reals"]}]}]]\ s\ t\ < \ 0))] $\endgroup$ Commented Jan 3, 2023 at 8:29
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    $\begingroup$ This is math, not Mathematica. At the present and in the near future ForAll and Exists do not deal with functions as variables. $\endgroup$ Commented Jan 3, 2023 at 9:56
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    $\begingroup$ Quantification over functions is second-order logic. Mathematica's theorem prover works only for a subset of first-order logic. However, it may be possible to answer this if you restrict your $f$ to a certain class of functions parametrized by a finite set of parameters, e.g a truncated Taylor series. $\endgroup$ Commented Jan 3, 2023 at 17:52
  • $\begingroup$ @flinty: I have strong doubts about trig polynomials instead of polynomials . $\endgroup$ Commented Jan 3, 2023 at 17:55