6:00 AM
There are two questions.
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How will be the Graph of the function $f(x,y)$ where $y=ϕ(x)$. From my intuition it seems it will not be a surface rather than a 3D curve! But I am not sure!

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What is the relation between two different functions, say $g(x)$ and $f(x)$, which have the same Legendre–Fenchel transformation $h(s)$? $$h(s) = \sup_{x\in I}\{sx - f(x)\} \quad \quad x \in I$$ h(s) = \sup_{x\in I'}\{sx - g(x)\} \quad \quad ...