Let f: R \rightarrow 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 x \in R. If f(0)=1, f^{'}(0)=2, and (\alpha, \beta) is the largest interval in which the function f (\log _{ e } x-x) is increasing, then \alpha+\beta is equal to \_\_\_\_ . [21 Jan 2026 Shift 1]

Application of Derivatives Part 3

Section: Mathematics

Question : 7 of 30

Marks: +1, -0

Let f:R⟶R be a twice differentiable function such that the quadratic equation f(x)m2−2f′(x)m+f′′(x)=0 in m , has two equal roots for every x∈R. If f(0)=1,f′(0)=2, and (α,β) is the largest interval in which the function f(logex−x) is increasing, then α+β is equal to ____ .

[21 Jan 2026 Shift 1]

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