Let `F:RtoR` be a thrice differntiable function. Suppose that `F(1)=0,F(3)=-4` and `F'(x)lt0` for all `x epsilon(1//2,3)`. Let `f(x)=xF(x)` for all `x inR`. Then the correct statement(s) is (are)

Options

A. a. `f'(1)lt0`
B. b. `f(2)lt0`
C. c. `f'(x)!=0` for an `x in (1,3)`
D. d. `f'(x)=0` for some `x in (1,3)`

Answer

A, B, C

Explanation

`f(x)=xF(x)`………….1
`:.f'(x)=xF'(x)+F(x)`…………….2
`impliesf'(1)=F'(1)+F(1)=F'(1)lt0`
`F(1)=0` and `F(3)=-4`
Also `F'(x)gt0` for all `xepsilon(1//2,3)`
So `F(x)` is decreasing and hence `F(2)lt0`.
`:.f(2)-2F(2)lt0`
Also for `x epsilon(1,3)`,
`f'(x)=xF'(x)+F(x)lt0`