Question

Let $F: R \rightarrow R$ 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 $x \in R$.
The correct statement(s) is(are)

• A. $f^{\prime}(1) < 0$
• B. $f(2) < 0$
• C. $ f^{\prime}(x) \neq 0$ for any $x \in(1,3)$
• D. $f^{\prime}(x)=0$ for some $x \in(1,3)$

Answer 1

Answer: (A), (B), (C)

$f^{\prime}(x)=x F^{\prime}(x)+F(x)$
$\Rightarrow f^{\prime}(1)=F^{\prime}(1)+F(1)=F^{\prime}(1)<0 \Rightarrow $ (1)
$f(2)=2 F(2)<0 \Rightarrow (2)$
for $x \in(1,3) f^{\prime}(x)=x f^{\prime}(x)+f(x)<0 \Rightarrow (3)$