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1)

Let f:[0,2]→  R  be a function which is continuous on [0,2] and is differentiable on (0,2)  with f(0)=1.Let   F(x)=x20f(t)dt,forx[0,2] . If  F'(x)= f '(x) for all  x ε (0,2) , then F(2) equal to


A) e21

B) e41

C) e-1

D) e4

Answer:

Option B

Explanation:

 Plan    Newton - Leibnitz.s formula

   ddx[ψ(X)ϕ(X)f(t)dt]=f{ψ(X)}{ddxψ(X)}f(ϕ(X)){ddxϕ(X)}

 Given, F(X)=x20f(t)dt

   F'(x)= 2x f(x)

 Also, F '(x)= f' (x)

    2xf(x)=f(x)f(x)f(x)=2x

  f(x)f(x)dx=2xdx

      ln f(x)= x2+ c

     f(x)=ex2+c

   f(x)=Kex2             [K=ec]

 Now, f(0)=1

                               1=K

 Hence     f(x)=ex2

       F(2)=40etdt

    =[et]40=e41