Mathematics Question from JEE Main 2026 · 21 January, Shift 1

Let $f: R \to R$ be a twice differentiable function such that the quadratic equation $f(x)m^2-2f'(x) m+f''(x) = 0$ in $m$, has two equal roots for every $x \in R$. If $f(0)=1, f'(0) = 2$, and $(\alpha, \beta)$ is the largest interval in which the function $g(x) = f(\log_e x - x)$ is increasing, then $\alpha+\beta$ is equal to: