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Q.

Let F:R→R be a thrice differentiable function. Suppose that F(1)=0,F(3)=−4  and F′(x)<0 for all x∈(1/2,3). Let f(x)=xF(x) for all x∈R If ∫13 x2F′(x)dx=−12 and ∫13 x3F′′(x)dx=40, then the correct expression (s) is (are)

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a

9f′(3)+f′(1)−32=0

b

∫13 f(x)dx=12

c

∫13 f(x)dx=−12

d

9f′(3)+f′(1)+32=0

answer is C.

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Detailed Solution

∫13 x3F′′(x)dx=40⇒x3F′(x)13−∫13 3x2F′(x)dx=40⇒x2F′(x)−xF(x)13−3(−12)=40⇒9f′(3)−3f(3)−f′(1)+f(1)=4⇒9f′(3)+36−f′(1)+0=4⇒9f′(3)−f′(1)+32=0⇒(C)⇒∫13 x2f′(x)dx=−12⇒x2f(x)13−∫13 2xf(x)dx=−12⇒−36−2∫13 f(x)dx=−12⇒∫13 f(x)dx=−12
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