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JEE Main202430 Jan 2024Morning ShiftMathematicsContinuity and DifferentiabilityActual

If the function f x = 1 | x | , | x | ≥ 2 a x 2 + 2 b , | x | < 2 is differentiable on R , then 48 ( a + b ) is equal to _______.

Correct answer

0

Step-by-step solution

Given, f x = 1 x , x ≥ 2 a x 2 + 2 b , - 2 < x < 2 - 1 x , x ≤ - 2 Also given f x is differentiable on R So, finding LHD and RHD at x = 2 we get, - 1 x 2 = 2 a x ⇒ - 1 4 = 4 a ⇒ a = - 1 16 Now, for function to be continuous at x = 2 we get, 1 2 = 4 a + 2 b ⇒ 1 2 = - 4 16 + 2 b ⇒ b = 3 8 Hence, 48 a + b = 48 3 8 - 1 16 ⇒ 48 a + b = 48 5 16 = 15

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