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Manipal MET2019MathematicsContinuity and Differentiability

Let f(x)=x^p ( 1 x ) , when x 0 and f(x)=0 , when x=0 . Then f(x) will be differentiable at x=0 , if

Options

  1. Ap 0
  2. Bp 1
  3. C0 p 1
  4. D1 2 p 1

Correct answer

B. p 1

Step-by-step solution

Given that, f(x)= aligned x^p ( 1 x ), & x 0 0, & x=0 aligned .f(x) will be differentiable at x=0 , if _ x 0 f(x)-f(0) x-0 exists finitely. _ x 0 x^p ( 1 x ) x exists finitely _ x 0 x^ p-1 ( 1 x ) exists finitely p-1 0 p 1

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