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1)

Let  f:RR and g:RR e two non-constant differentiable functions. If  f(x)=(e(f(x)g(x))g(x) for all xR  and f(1)=g(2)=1, then which of the following statement(s) is (are) TRUE ?


A) f(2)<1loge2

B) f(2)>1loge2

C) g(1)>1loge2

D) g(1)<1loge2

Answer:

Option B,C

Explanation:

We have,

    f(x)=e(f(x)g(x))g(x)xR

        f(x)=ef(x)eg(x)g(x)

     f(x)ef(x)=g(x)eg(x)

    ef(x)f(x)=eg(x)g(x)

  On integrating both sides, we get

   ef(x)=eg(x)+c

  At x=1,

  ef(1)=eg(1)+c

   e1=eg(1)+c    [  f(1)= 1]  ......(i)

At x=2,

 ef(2)=eg(2)+c

    ef(2)=e1+C  [    g(2)= 1]    ..........(ii)

 From Eqs. (i)   and (ii)

 ef(2)=2e1eg(1)              ............(iii)

       ef(2)>2e1

 We know that , e-x  is decreasing

      f(2)<loge21

                   f(2)>1loge2

    eg(1)+ef(2)=2e1    [from Eq,(iii)]

eg(1)<2e1

    g(1)<loge21

     g(1)>loge2