NTA Abhyas JEE Main2020MathematicsContinuity and DifferentiabilityPractice

If f x = 2 x 2 + 3 ; x > 3 a x 2 + b x + 1 ; x ≤ 3 is differentiable everywhere, then a b 2 is equal to

Options

A5
B7 3
C1
D16 9

Correct answer

C. 1

Step-by-step solution

∵ f x is continuous at x = 3 ∴ l i m x → 3 f x = f 3 2 3 2 + 3 = a 3 2 + b 3 + 1 9 a + 3 b = 20 … 1 Now, f ' x = 4 x ; x > 3 2 a x + b ; x ≤ 3 ∵ f x is also differentiable at x = 3 ∴ 4 3 = 2 a 3 + b 6 a + b = 12 … 2 From 1 & 2 , we get, 9 a = 16 ⇒ a = 16 9 ,   b = 4 3

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