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I need to solve the following equation for $x$ in [0, 1]. Assume $0<\alpha<1$ and $0<\lambda$.

$$(1 - x)^{\alpha+1} - \lambda (x+1)^{\alpha+1} = -2\lambda (\alpha + 1) x^\alpha$$

Would very much appreciate any kind of help!

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Let

$$y(x,\lambda,\alpha)=2\lambda(\alpha+1)x^\alpha+(1-x)^{\alpha+1}-\lambda(1+x)^{\alpha+1}$$

be the function we are finding roots of, writing $y=y(x)=y(x,\lambda,\alpha)$ for shorthand. Then notice our boundary cases are

$$y(0)=1-\lambda\implies\begin{cases}y(0)\le0,&\lambda\ge1\\y(0)>0,&\lambda<1\end{cases}$$ $$y(1)=2\lambda(\alpha+1-2^\alpha)\ge0$$

which means that when $\lambda\ge1$, the uniroot function finds the root with guaranteed convergence because $y(0)\le0\le y(1)$ has a root in-between.

And although I haven't proven it, graphing strongly suggests that there are no solutions for $0<\lambda<1$, so there is no point in root-finding there.

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