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I am stuck on the following problem:

Given $ f(x)=\displaystyle\frac{4x+3}{x^2+1}$, find how many roots the equation $$f(f(x))=\int_3^4{f(x)\mathrm{d}x}$$ has in the interval $[1, 4]$.

Consulting GeoGebra for the graphs, there is a root in that interval. Obviously, I first tried to use the minimum/maximum on $f(x)$ in the interval and the monotonicity so as to restrict $f(f(x))$ and the integral to a common interval (The integral's interval should be a subinterval of the function's interval so the existence and uniqueness of the root would be implied by the intermediate value theorem and the function's monotonicity - it is strictly increasing.) but that hasn't worked out. I also computed the integral to be $\ln{\frac{289}{100}}+3\cot^{-1}{13}\approx 1.29$ and then $f(f(1))\approx 1.28< 1.29 < 3.32 = f(f(4))$ but these are based on approximations while I would want a not-so-computational (theoretical, as you may call it) approach, while of course leaving room for some necessary but reasonable computations (not that I would have to approximate $cot^{-1} !)$. Thanks in advance.

Peter Allen
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  • A root will be a root of the 4x+3. So $x=-3/4$. No? – user29999 May 05 '23 at 18:25
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    @user29999 that does not reconcile with the right-hand side of OP's equation. – Randall May 05 '23 at 18:33
  • I think your argument is a good start and is what I would try, too. By continuity you know there will be one solution and by monotonicity there can be only one. – Randall May 05 '23 at 18:34
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    The definite integral is just a number C. What you have to solve is $f(y)=C\Leftrightarrow \frac{4y+3}{y^2+1}=C$. This is a quadratic in y with at most 2 roots. Once you’ve got y you could find the x. – WindSoul May 05 '23 at 18:34
  • FWIW, if I did the algebra correctly, the LHS expands to $\frac{3x^4 + 16x^3 + 18x^2 + 16x + 15}{x^4 + 18x^2 + 24x + 10}$. – Dan May 05 '23 at 20:06

1 Answers1

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This problem solves numerically.

  1. Calculate the definite integral $C=\int_3^4{\frac{4x+3}{1+x^2}dx}$ to approximately $C\approx 1.2916$
  2. Solve $f(y)=C\Rightarrow y\approx -0.38038\text{ or } y\approx 3.4773$
  3. Solve $f(x)=y$ for each value of y obtained at previous step
  4. Out of 4 solutions, only one is within $[1,4]: f(x)=3.4773, x\approx 1.015$
WindSoul
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