Let f : a ,   b → 1 ,   ∞ be a continuous function and let g   : R → R be…

Let f:a, b→1, ∞ be a continuous function and let g :R→R be defined as
gx= 0ifx<a,∫axftdtifa ≤x ≤b,∫abftdtifx>b . Then
  1. gx is continuous but not differentiable at a
  2. gx is differentiable on R
  3. gx is continuous but not differentiable at b
  4. gx is continuous and differentiable at either a or b but not both

Solution

Since fx ≥1 ∀ x ∈a, b for gx
LHD at x=a is zero
And RHD at x=a= limx→a+⁡∫axftdt-0x-a= limx→a+⁡fx ≥1
Hence gx is not differentiable at x=a
Similarly LHD at x=b is greater than 1
gx is not differentiable at x=b

Asked in: JEE Advanced 2014 (Paper 1)

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