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1)

 Let   F:RR  be a thrice  differentiable  function. Suppose that  F(1)=0,F(3)=-4  and F'(x)<0 for x ε (1,3)  , Let f(x)=xF(X) for all x ε R.

The correct statement(s)  is/are


A) f1(1)<0

B) f(2)<0

C) f(x)0 for any xϵ(1,3)

D) f'(x) =0 for same xϵ(1,3)

Answer:

Option A,B,C

Explanation:

According to the given data,

   F(x)<0,xϵ(1,3)

 We have, f(x)= x F(x)

            f(x)=F(x)+xF(x)        .......(i)

      f(1)=F(1)+F(1)<0

         [given F(1)=0 and F'(x)<0]

 Also, f(2)=2F(2)<0

                   [using   F(x)<0,xϵ(1,3)  ]

Now,  f'(x)=F(x)+x F'(x)<0

                 [using   F(x)<0,xϵ(1,3)  ]

       f(x)<0