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JEE Advanced2010MathematicsContinuity and DifferentiabilityActual

Let f be a real-valued function defined on the interval (0, ) , by f(x)= x+ ₀^x 1+ t d t . Then which of the following statement(s) is (are) true ?

Options

  1. Af^ (x) exists for all x (0, )
  2. Bf^ (x) exists for all x (0, ) and f^ is continuous on (0, ) , but not differentiable on (0, )
  3. Cthere exists >1 such that |f^ (x) | < |f(x)| for all x ( , )
  4. Dthere exists >0 such that |f(x)|+ |f^ (x) | from all x (0, )

Correct answer

B. f^ (x) exists for all x (0, ) and f^ is continuous on (0, ) , but not differentiable on (0, )

Step-by-step solution

Here, f^ (x)= 1 x + 1+ x , x>0 but f(x) is not differentiable in (0, ) as x may be -1 and then f^ (x)=- 1 x^2 + x 2 1+ x will not exists. f^ (x) is continuous for all x (0, ) but f^ (x) is not differentiable on (0, ) . Option (b) is true. Also, f^ (x) 3 , if x>1 and f(x)>3 , if x>e^3 Let =e^3 Option (c) is true. (d) is not possible as f(x) when x . Hence, (b, c) is the correct option.

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