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Prove (or disprove): If $a,b,c,d$ are positive real numbers with $a\times b=c\times d$, then the only solutions for $x$ in the equation $$\frac {a^x+b^x}{c^x+d^x} = \frac{a+b}{c+d}$$ are $x = \pm 1$.

Other than the obvious $a=b=c=d$ solution.

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  • $\begingroup$ Consider taking the limit as $x \rightarrow \pm \infty$. This would prove the only solution is $\pm1$ $\endgroup$
    – MathsPro
    Apr 19, 2015 at 14:22
  • $\begingroup$ Can you elaborate on it MathsPro? $\endgroup$
    – GohP.iHan
    Apr 19, 2015 at 16:58

1 Answer 1

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Clearly, if $a = c, b = d$ or $a = d, b = c$, then all $x$ satisfy the equation. Hence the statement is false. However, one can show that this is the only exception, namely, if $a,b,c,d > 0$ such that $ab = cd$ and $\{a,b\}\neq\{c,d\}$, then $x = \pm 1$ is the only solution to $$\frac{a^x+b^x}{a+b} = \frac{c^x+d^x}{c+d}.$$

Suppose $ab = cd = s^2$. Without loss of generality, we may assume $s = 1$ since otherwise we can divide $a,b,c,d$ by $s$ and the solution set will not change. We may also assume $a \ge b$ and $c \ge d$. Now $b = 1/a, d = 1/c$ and $a,c \ge 1$.

Consider $f\colon [0,\infty)\times \mathbb{R}\to\mathbb{R}$ defined by $$f(z, x) = \frac{e^{zx} + e^{-zx}}{e^{z} + e^{-z}}.$$

Claim $f(z,x)$ is decreasing in $z$ if $|x| < 1$ and is increasing in $z$ if $|x| > 1$. Therefore $f$ is injective in $z$ if $x\neq \pm 1$.

Suppose the claim holds. We know that $f(\log a,x)=f(\log c,x)$ implies either $x = \pm 1$ or $a = c$.

Proof of Claim Observe that $$\partial_z f(z,x) = (-1 + x)\left(e^{(1+x)z}-e^{-(1+x)z}\right) + (1+x)\left(e^{(x-1)z}-e^{(1-x)z}\right).$$ If $|x| < 1$, then all terms above are negative and so $f$ is decreasing in $z$. If $|x| > 1$, then all terms above are positive, and so $f$ is increasing in $z$.

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