Let f: R → R be a twice differentiable function such that the quadratic equation f(x) m^2-2 f^′(x) m +f^′ ′(x)=0 in m, has two equal roots for every…

Let $\displaystyle f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that the quadratic equation $\displaystyle f(x) \mathrm{m}^2-2 f^{\prime}(x) \mathrm{m}+f^{\prime \prime}(x)=0$ in m, has two equal roots for every $\displaystyle x \in \mathbf{R}$. If $\displaystyle f(0)=1, f^{\prime}(0)=2$, and ( $\displaystyle \alpha, \beta$ ) is the largest interval in which the function $\displaystyle f\left(\log _{\mathrm{e}} x-x\right)$ is increasing, then $\displaystyle \alpha+\beta$ is equal to $\displaystyle \_\_\_\_$.