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Let F:RR to RR be a thrice differentiabl...

Let `F:RR to RR` be a thrice differentiable function. Suppose that `F(1)=0,F(3)=-4` and `F'(x)lt0` for all `x in((1)/(2),3)`. Let `f(x)=xF(x)` for all `x in RR`.
The correct statement(s) is (are)-

A

`f'(1)lt0`

B

`f(2)lt0`

C

`f'(x)ne0` for any `x in(1,3)`

D

`f'(x)=0` for some `x in(1,3)`

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