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What is the easiest way to compute the convex conjugagte of a real convex function $f: \mathbb{R} \to \mathbb{R}$, defined by $f^*(s) = \sup_{x} \{ s x - f(x) \}$

I know I can compute the derivative of the inner objective function w.r.t $x$, solve the equation comparing the derivative to 0, and substitute the result into the objective to discover $f^*$. However, is there a built-in function, or an easy one-liner to do that?

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    $\begingroup$ fc[s_] := MaxValue[s x - f[x], x]? $\endgroup$
    – Michael E2
    Commented Dec 14, 2019 at 23:16
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    $\begingroup$ @MichaelE2, actually, it does not work for very simple cases, i.e. MaValue[s x - Log[1+E^x], x] says that no global maxima found (although clearly the function is strictly concave and the unique critical point is a global maximum). $\endgroup$
    – Alex Shtoff
    Commented Dec 16, 2019 at 9:19
  • $\begingroup$ @AlexShtoff Michael is right $\endgroup$
    – yode
    Commented Feb 23 at 23:59

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