Let a ,   b ∈ R and f   :   R → R be defined by f x = a cos ⁡ x 3 - x + b x…

Let a, b∈R and f : R→R be defined by fx=acos⁡x3-x+bxsin⁡x3+x . Then f is -
  1. Differentiable at x=0 if a=0 and b=1
  2. Differentiable at x=1 if a=1 and b=0
  3. NOT differentiable at x=0 if a=1 and b=0
  4. NOT differentiable at x=1 if a=1 and b=1

Solution

If x3-x ≥0    ⇒ cos⁡x3-x=cos⁡x3-x
x3-x<0   ⇒  cos⁡x3-x=cos⁡x3-x
Similarly bxsin⁡x3+x=bxsin⁡x3+x for all x∈R
∴ fx=acos⁡x3-x+bxsin⁡x3+x
Which is composition and sum of differentiable functions therefore always continuous and differentiable.

Asked in: JEE Advanced 2016 (Paper 2)

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