Let $f(x)=\frac{x}{e^x}$, and $f(a)=f(b), a<1<b$. How to prove that $\frac{a^2}{a-1}+\frac{b^2}{b-1}>\frac{10}{3}$? I think to use a derivative but I don't know how?
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Where does that problem come from? – Martin R Jun 12 '22 at 11:40
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It's similar to this: https://math.stackexchange.com/questions/2792428/let-fx-x-1-ln-x-and-given-0-a-b-if-fa-fb-prove-that-f – piteer Jun 12 '22 at 12:31
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@piteer We need to prove that $0 < a < g(b)$ (from the inequality to be proved). It suffices to prove that $f(g(b)) > f(b)$ for all $b > 1$. But it is complicated (even by a computer). – River Li Jun 12 '22 at 23:23