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JEE Main2014MathematicsContinuity and DifferentiabilityActual

Let f , g : R R be two functions defined by f(x)= cases x ( 1 x ) & , x 0 0, & , x=0 cases and g(x)=x f(x) Statement I: f is a continuous function at x =0 . Statement II: g is a differentiable function at x=0 .

Options

  1. ABoth statement I and II are false.
  2. BBoth statement I and II are true.
  3. CStatement I is true, statement II is false.
  4. DStatement I is false, statement II is true.

Correct answer

B. Both statement I and II are true.

Step-by-step solution

aligned &f(x)= cases x ( 1 x ), & x 0 0, & x=0 cases & and g(x)=x f(x) & For f(x) & LHL = _ h 0⁻ -h (- 1 h ) aligned =0 a finite quantity between -1 and 1 =0 RHL = _ h 0⁺ h 1 h =0 Also, f(0)=0 Thus LHL = RHL =f(0) f(x) is continuous at x=0 g(x)= cases x^2 1 x , & x 0 0, & x=0 cases For g(x) LHL = _ h 0⁻ -h^2 ( 1 h ) =0^2 a finite quantity between -1 and 1=0 RHL = _ h 0⁺ h^2 ( 1 h )=0 Also g(0)=0 g(x) is continuous at x=0

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