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Let f(x)= (x^3, x^2 ) and g(x)=[x]^2+ x ^2 , where [x] denotes the greatest integer and x denotes the fractional part function. Then which of the following holds?

Options

  1. Af is continuous for all x
  2. Bg is discontinuous for all x I
  3. Cf is differentiable for all x (1, )
  4. Dg is not differentiable for all x I

Correct answer

C. f is differentiable for all x (1, )

Step-by-step solution

f ( x )= [ array ll x ^3 & if x 1 x ^2 & if x 1 array ; g ( x )= [ array ll x +2 & -1 x 0 x & 0 x 2 x +2 & 2 x 3 array . . also g(x) be continuous at x I . g(n)=n^2 ; g (n⁺ )=n^2 ; g (n⁻ )=(n-1)^2+1 for continuity n^2=(n-1)^2+1 n =1 g(x) is continuous only at one integer x=1 Note f is obviously continuous for all x and derivable for all x (1, )]

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