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Let f, g: R R be functions defined by f(x)= array cc x ( 1 x ), & for x 0 0, & for x=0 array . and g(x)=x f(x) Consider the following statements (i) f(x) is continuous at x=0 but not differentiable at x=0 (ii) g(x) is differentiable at x=0 , but g^1(x) is not continuous at x=0 Then, which one of the following is true?

Options

  1. A(i) is true; but (ii) is false
  2. BBoth (i) and (ii) are true
  3. C(i) is false, but (ii) is true
  4. DBoth (i) and (ii) are false

Correct answer

B. Both (i) and (ii) are true

Step-by-step solution

We have, aligned & f(x)= array cc x ( 1 x ), & for x 0 0, & for x=0 array . & g(x)=x f(x) aligned _ x 0 f(x)= _ x 0 x ( 1 x )=0 1 x [-1,1] f(x) is continuous at x=0f^ (x)= array cc ( 1 x )- 1 x ( 1 x ), & x 0 0, & x=0 array . f(x) is not differentiable at x=0

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