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JEE Advanced2010MathematicsDifferentiationActual

Let f be a real-valued function defined on the interval (-1,1) such that e^ -x f(x)=2+ ₀^x t^4+1 d t, for all x (-1,1) and let f⁻¹ be the inverse function of f . Then (f⁻¹ )^ (2) is equal to

Options

  1. A1
  2. B1 3
  3. C1 2
  4. D1 e

Correct answer

B. 1 3

Step-by-step solution

We have, e^ -x f(x)=2+ ₀^x t^4+1 d t x (-1,1) On differentiating w.r.t. x , we get array ll & e^ -x (f^ (x)-f(x) )= x^4+1 & f^ (x)=f(x)+ x^4+1 e^x & f⁻¹ is the inverse of f & f⁻¹(f(x))=x & f^ -1^ (f(x)) f^ (x)=1 & f^ -1^ (f(x))= 1 f^ (x) & f^ -1^ (f(x))= 1 f(x)+ x^4+1 e^x At & x=0, f(x)=2 & f^ -1^ (2)= 1 2+1 = 1 3 array

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