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JEE Advanced2008MathematicsContinuity and DifferentiabilityActual

Let f and g be real valued functions defined on interval (-1,1) such that g^ (x) is continuous, g(0) 0, g^ (0)=0, g^ (0) 0 and f(x)=g(x) x . Statement 1 _ x 0 [g(x) x-g(0) cosec x]=f^ (0) . Statement 2 f^ (0)=g(0) .

Options

  1. AStatement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
  2. BStatement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
  3. CStatement 1 is true, Statement 2 is false.
  4. DStatement 1 is false, Statement 2 is true

Correct answer

B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

Step-by-step solution

We have, aligned & _ x 0 g(x) x-g(0) x & = _ x 0 g^ (x) x-g(x) x x =0 aligned Since, f(x)=g(x) x aligned & f^ (x)=g^ (x) x+g(x) x & f^ (x)=g^ (x) x+2 g^ (x) x-g(x) x & f^ (0)=0 aligned Thus, _ x 0 [g(x) x-g(0) cosec x]=0=f^ (0) Statement 1 is true. Statement 2. f^ (x)=g^ (x) x+g(x) x f^ (0)=g(0) Statement 2 is not a correct explanation of Statement 1.

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