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NDA2016MathematicsContinuity and DifferentiabilityActual

Paragraph: Let f ( x )= array ll -2, & -3 x 0 x -2, & 0 x 3 array . and g ( x )= f (| x |)+| f ( x )| Question: Which of the following statements are correct? 1. g ( x ) is continuous at x =0 . 2. g(x) is continuous at x=2 . 3. g ( x ) is continuous at x =-1 . Select the correct answer using the code given below:

Options

  1. A1 and 2 only
  2. B2 and 3 only
  3. C1 and 3 only
  4. D1,2 and 3

Correct answer

D. 1,2 and 3

Step-by-step solution

At x=0 For LHL: g(x)=-2+|-2|=0 For RHL: g(x)=|x|-2+|x-2| aligned g(x) & =x-2-(x-2)=0 g(x) & =0 aligned For (x=0): g(x)=-2+|-2|=0 aligned & LHL = _ x 0⁻ g(x)=0 & RHL = _ x 0⁺ g(x)=0 & g(0)=0 aligned g(x) is continuous at x=0 At x=2 For LHL: g(x)=|x|-2+|x-2| aligned & g(x)=x-2-(x-2) & g(x)=0 aligned For RHL: g(x)=|x|-2+|x-2| aligned & g(x)=x-2+x-2 & 2(x)=2 x-4 aligned For (x=2): g(x)=|x|-2+|x-2| LHL = _ x 2⁻ g(x)=0 aligned & RHL = _ x 2⁺ g(x)= _ x 2 2 x-4=2(2)-4=0 & g(2)=|2|-2+|2-2| & g(2)=2-2+2-2=0 aligned g(x) is c

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