Let F   :   R → R be a thrice differentiable function. Suppose that F 1 = 0 ,   F 3 =…

Let F : R→R be a thrice differentiable function. Suppose that F1=0, F3= -4  and F′x<0 all x∈12, 3. Let fx=xF(x) for all x∈ R .
If ∫13x2F′xdx= -12  and ∫13x3 F′′xdx=40, then the correct expression(s) is(are)
  1. 9f′3+f′1-32=0
  2. ∫13fxdx=12
  3. 9f′3-f′1+32=0
  4. ∫13fxdx= -12

Solution

∫13x2F′xdx=-12 and ∫13x3 F′′xdx=40
∫13x3F′′x=x3F′x]13-∫133x2F′(x)
40=27F′3-F′1+36
4=9 f3+4-f(1)
⇒  9f′3-f′1+32=0
Also
∫13x2F′(x)dx
=x2Fx]13-∫132xFxdx=-12
9F3-F1-2∫13xFxdx=-12
-36-0-2∫13fxdx=-12
-36+12=2∫13f(x)dx
-12=∫13f(x)dx

Asked in: JEE Advanced 2015 (Paper 2)

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