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JEE Advanced2016MathematicsContinuity and DifferentiabilityActual

Let f : - 1 2 , 2 → R and g : - 1 2 , 2 → R be functions defined by f x = x 2 - 3 and g x = x f x + 4 x - 7 f x , where y denotes the greatest integer less than or equal to y for y ∈ R . Then

Options

  1. Af is discontinuous exactly at three points in - 1 2 , 2
  2. Bf is discontinuous exactly at four points in - 1 2 , 2
  3. Cg is NOT differentiable exactly at four points in - 1 2 , 2
  4. Dg is NOT differentiable exactly at five points in - 1 2 , 2

Correct answer

B. f is discontinuous exactly at four points in - 1 2 , 2

Step-by-step solution

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

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