Question on intersection of two curves and inverses

Question

Maybe the question is not very well posed.

So, I have a say strictly increasing, continuous, convex function $f$. Suppose that $f(1) =0$, and I force the domain of $f$ to be say $[1,2]$. Then there is only solution to the system $f(x_1)=f(x_2)$, which is $x_1=x_2$ other than $x_1=x_2=1$. Now, I have $a,b>0$ then what can I say about the solution of $af(x_1)=bf(x_2)$? Feel free to make any assumptions on $f$, but present those, under which I can say anything about the relation between $x_1,x_2$.

Answer 1

The solution is

$$x_2=f^{-1}\left(\frac abf(x_1)\right)$$ which is "some function", continuous and monotonic.

---

For example, $f(x)=x^k$ yields

$$x_2=\sqrt[k]{\frac ab}x_1.$$

$f(x)=\sinh x$ yields

$$x_2=\text{argsinh}\left(\frac ab\sinh(x_1)\right).$$

Answer 2

@YvesDaoust can I kindly ask why you deleted your post on that thread I asked, I never offended you - merely wanted to understand your answer better in future. Can you please repost it? I am posting here since I can't message you for some reason. p.s. Ik ben geboren in Gruitrode :P