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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 is/are correct? 1. g(x) is differentiable at x=0 . 2. g ( x ) is differentiable at x =2 . Select the correct answer using the code given below:

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

D. Neither 1 nor 2

Step-by-step solution

aligned & f(x)= array cc -2, & -3 x 0 x-2, & 0 x 3 array . and & g(x)=f(|x|)+|f(x)| & array l At x=0 array & For LHD : g(x)=-2+|-2|=-2+2=0 g(x)=0 & LHD = _ x 0⁻ g(x)-g(0) x-0 = _ h 0 g(-h)-g(0) -h & = _ h 0 0-0 -h = _ h 0 0 & LHD =0 aligned For RHD : g(x)=|x|-2+|x-2|g(x)=x-2-(x-2) x 0 (and just greater than zero) g(x)=x-2-x+2=0 Now g ( x ) is not continuous at x=0 , hence g(x) is not differentiable at x=30 At x=2 For LHD : aligned g(x) & =|x|-2+|x-2|=x-2-(x-2) & =x-2-x+2=0 aligned LHD = _ x 2⁻ g(x)-g(2) x-2 = _ x 2

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