Paragraph: Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be a thrice differentiable function. Suppose that…

Paragraph: Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be a thrice differentiable function. Suppose that $F(1)=0, F(3)=-4$ and $F^{\prime}(x)<0$ for all $x \in(1 / 2,3)$. Let $f(x)=x F(x)$ for all $x \in \mathbb{R}$.
Question: The correct statement(s) is(are)
  1. f′1<0
  2.  f2<0
  3. f′x≠0 for any x∈(1, 3)
  4. f′x=0 for some x∈(1, 3)

Solution

F1=0, F3= -4
F′x<0
F(x) is decreasing, also Fx<0
Now, fx=xF(x)
x∈12, 3 xFx<0
∴   f2<0
Also,
f′x=xF′x+F(x)
f′1=F′1<0
Also, f(x) is strictly decreasing
Now, F1=Fx=0
∴   f′x≠0   as F′x<0  and fx is decreasing.

Asked in: JEE Advanced 2015 (Paper 2)

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