I'd like to find a method to solve the problem $$\min \Biggl\{ \frac{x^2+b_1x+c_1}{x^2+b_2x+c_2}:\quad 0\leq x\leq \bar{x} \Biggr\}$$ where the variable $x\in \mathbb{R}$ is bounded on $[0,\bar{x}]$ and the quotient is supposed to be well defined on this box. It seems that we should be able to find a closed-form solution by discussing the coefficients. Hoping to get an answer to this question. Thanks in advance!
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Share what you tried. – Anne Bauval Jan 05 '23 at 19:01
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1@AnneBauval I tried the one in my answer. Hope it is fine, and welcome any better solution. – kaienfr Jan 05 '23 at 20:27
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I've just figured out a way which maybe a best one as follows: Let $$\varphi(x) = \frac{x^2+b_1x+c_1}{x^2+b_2x+c_2}$$ whose derivative is $$\varphi'(x) = \frac{(b_2-b_1)x^2 + 2(c_2-c_1)x + b_1c_2-c_1b_2}{(x^2+b_2x + c_2)^2}.$$ Clearly, $\varphi'$ is well defined on $[0,\bar{x}]$ since the denominator never being zero on this box (by assumption).
Hence, all roots of $\varphi'$ are exactly the roots of the polynomial $$(b_2-b_1)x^2 + 2(c_2-c_1)x + b_1c_2-c_1b_2,$$ which can be very easily computed by roots formula.
Now, let $Z$ be all real roots of this polynomial containing in the box $[0,\bar{x}]$, and let $$C = \{0,\bar{x}\}\cup Z.$$ Clearly, all minimum of $\varphi$ over $[0,\bar{x}]$ must be included in $C$. So we just need to check all values of $\varphi$ on $C$ (with at most $4$ different values) and take their minima.
kaienfr
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