NTA Abhyas JEE Main2020MathematicsContinuity and DifferentiabilityPractice

Let g x = x f x , where f x = x 2 sin ⁡ 1 x : x ≠ 0 0 : x = 0 . At x = 0 ,

Options

Ag is differentiable but g ′ is not continuous
Bg is differentiable while f is not differentiable
Cboth f and g are non differentiable
Dg is differentiable and g ′ is continuous

Correct answer

D. g is differentiable and g ′ is continuous

Step-by-step solution

We have, g x = x 3 sin ⁡ 1 x : x ≠ 0 0 : x = 0 For x ≠ 0 , g ′ x = x 3 cos 1 x − 1 x 2 + 3 x 2 sin 1 x = - x cos ⁡ 1 x + + 3 x 2 sin ⁡ 1 x For x = 0 , g ' 0 = lim x → 0 g x − g 0 x − 0 = lim x → 0 x 3 sin 1 x − 0 x = lim x → 0 x 2 sin ⁡ 1 x = 0 . ∴ g ' x = 3 x 2 sin 1 x − x cos 1 x ,   x ≠ 0 0 ,   x = 0 g ′ is continuous at x = 0 Also, g is differentiable at x = 0 .

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