Let f   : - 1 2 ,   2 → R and g : - 1 2 , 2 → R be functions defined by f x = x 2 - 3…

Let f :-12, 2→R and g:-12,2→R be functions defined by fx=x2-3 and gx=xfx+4x-7fx , where y denotes the greatest integer less than or equal to y for y∈R . Then
  1. f is discontinuous exactly at three points in -12, 2
     
  2. f is discontinuous exactly at four points in -12, 2
     
  3. g is NOT differentiable exactly at four points in -12, 2
  4. g is NOT differentiable exactly at five points in -12, 2

Solution

fx=x2-3
gx=x+4x-7 x2-3
∵ f is discontinuous at (when x2 is an integer)
x=1, 2, 3, 2 in -12, 2
and x+4x-7≠0 at x=1, 2, 3, 2
⇒ gx is discontinuous at x=1, 2, 3  in -12, 2
In 0-δ, 0+δ       ( δ is very small)
⇒ gx=x+4x-7. -3
⇒ 'g' is non derivable at x=0 .as | x | is there which is not diff. at x=0
In 74-δ,74+δ
⇒ gx=0 as fx=0
⇒ Derivable at x=74
∴ 'g' is non - derivable at 0, 1, 2,3

Asked in: JEE Advanced 2016 (Paper 2)

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