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I have the following function

$$ f(x) = \frac{1+a\sqrt{bx-1}}{1+a\sqrt{cx-1}} $$ on $[1, \infty)$.

$$ a > 0 \quad b, c\ge 1$$

I would like to find an upper bound on this function, one that is tight when $b = c$.

I have tried various transformations, looking at the derivative, but to no avail.

My hope was that the extrema are at $x = 1$ and $x = \infty$, but numerical simulations contradict this.

Any help would be appreciated.

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  • $\begingroup$ Yeah, $f(x)$ has either a minima or a maxima over $(0,\infty)$, depending on the relative magnitude of $b$ and $c$ . It'd take a lot of work, though. $\endgroup$ – Quanto Oct 31 '19 at 17:36

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