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Let F:RrarrR be a thrice differentiable ...

Let `F:RrarrR` 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 R`.
If `int_1^3x^2F'(x)dx=-12` and `int_1^3x^3F''(x)dx=40`, then the correct expression(s) is (are)

A

`9f'(3)+f'(1)-32=0`

B

`int_(1)^(3)f(x)dx=12`

C

`9f'(3)-f'(1)+32=0`

D

`int_(1)^(3)f(x)dx=-12`

Text Solution

Verified by Experts

The correct Answer is:
C, D
`int_(1)^(3)x^(3)F''(x)dx=40`
`implies [x^(3)F'(x)]_(1)^(3)-int_(1)^(3)3x^(2)F'(x)dx=40`
`implies [x^(2)f'(x)-xf(x)]_(1)^(3)-3(-12)=40` (Using 1 and 2)
`implies9f'(3)-3f(3)-f'(1)+f(1)=4`
`implies9f'(3)+36-f'(1)+0=4 ( :' F(1),:.f(1=0))`
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