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Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Consider a differentiable function f: R R with f(0)=0 and f^ (0)=1 . Which of the following statements are true for any such function f ?

Options

  1. Af(x)>0 on (0, q) for some positive q .
  2. Bf(x) is increasing on (p, q) for some negative p and some positive q .
  3. CThere exists a differentiable function g: R R such that g^ (x)=f(x) and
  4. Df^ (x) is continuous.

Correct answer

C. There exists a differentiable function g: R R such that g^ (x)=f(x) and

Step-by-step solution

Consider a differentiable function f: R R such that f(0)=0 and f^ (0)=1 . (b) f(x)=x-2 x^2 ( 1 x ) , if x is distinct to 0 and f(0)=0 . (d) h(x)= ₀^x f(t) d t fulfills h^ (x)=f(x) . g(x)= ₀^x h(t) d t g^ (x)=h(x) g^ (x)=h^ (x)=f(x)

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