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$$f(x) = \frac{\sqrt{x}\int_a^1 e^{-xs} s^{b+1}~ \mathrm{d}s}{\int_a^1 e^{-xs} s^{b} ~\mathrm{d}s}:\left]0,+\infty\right[\ \to \mathbb{R} $$

where $0 < a < 1$ and $b > 0.$

By applying elementary rules of differentiation I can only prove that the function is monotonically increasing in $\left]\ 0,\frac{1}{1-a}\ \right]$

But I need to prove that it is monotonically increasing in $\left]\ 0,+\infty\ \right[$

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Your conjecture is incorrect. In particular, for $a=b=1/10$ we have $f'(5)=-0.00561\ldots<0$.

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