If f x = x p cos ⁡ 1 x , x ≠ 0 0 , x = 0 is differentiable at x = 0 , then

Question

If f x = x p cos ⁡ 1 x , x ≠ 0 0 , x = 0 is differentiable at x = 0 , then, p 0, 0 p 1, p = 1, p > 1

MARKS App · Accepted Answer

Given, f x = x p cos ⁡ 1 x , x ≠ 0 0 , x = 0 Since, f ( x ) is differentiable at x = 0 , therefore it is continuous at x = 0 ∴ l i m x → 0 f x = f 0 = 0 ⇒ l i m x → 0 x p cos ⁡ 1 x = 0 ⇒ p > 0 As f x is differentiable at x = 0 ∴ l i m x → 0 f x - f ( 0 ) x - 0 exists finitely ⇒ l i m x → 0 x p cos ⁡ 1 x - 0 x exists finitely ⇒ l i m x → 0 x p - 1 cos ⁡ 1 x exists finitely ⇒ p - 1 > 0 ⇒ p > 1

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