JEE Main202624 January 2026Morning ShiftMathematicsApplication of DerivativesActual

Let (2 , ) be the largest interval in which the function f(t)= |t+1| t² , t 2 , is _ _ _ _

Correct answer

Step-by-step solution

For t For -1 f(t) = t+1 t^2 , so f'(t) = -(t+2) t^3 > 0 (increasing). For t f(t) = -t-1 t^2 , so f'(t) = t+2 t^3 . For -2 f'(t) For t f'(t) > 0 (increasing). Largest decreasing interval is (-2, -1) = (2 , ) , so = -1 . For g(x) = 2 (x-2) - x^2 + 4x + 1 : g'(x) = 2 x-2 - 2x + 4 = 0 gives (x-2)^2 = 1 , so x = 3 . Since g''(x) = - 2 (x-2)^2 - 2 g(3) = 2 (1) - 9 + 12 + 1 = 4

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