Discussion Forum



1)

 Let   $F:R\rightarrow R$  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) $f^{1}(1)<0$

B) $f(2)<0$

C) $ f'(x)\neq 0$ for any $x\epsilon (1,3)$

D) f'(x) =0 for same $x\epsilon (1,3)$

Answer:

Option A,B,C

Explanation:

According to the given data,

   $F(x)<0,\forall x\epsilon(1,3)$

 We have, f(x)= x F(x)

$\Rightarrow$            $f'(x)=F(x)+xF'(x)$        .......(i)

$\Rightarrow$      $ f'(1)=F(1)+F'(1)<0$

         [given F(1)=0 and F'(x)<0]

 Also, f(2)=2F(2)<0

                   [using   $F(x)<0,\forall x \epsilon (1,3)$  ]

Now,  f'(x)=F(x)+x F'(x)<0

                 [using   $F(x)<0,\forall x \epsilon (1,3)$  ]

       $\Rightarrow f'(x)<0$