Question

The correct statements is (are)

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

Answer 1

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

The correct options are
A $f^{\prime }\left(1\right)<0$
B $f^{\prime }\left(x\right)\ne 0$ for any $x\in (1,3)$
C $f\left(2\right)<0$

$f\left(x\right)=xF\left(x\right)$
Differentiating both side w.r.t $x$
$f^{\prime }\left(x\right)=F\left(x\right)+x{F}^{\prime }\left(x\right)$
$\therefore f^{\prime }\left(1\right)=F\left(1\right)+F^{\prime }\left(1\right)=0+F^{\prime }\left(1\right)<0\because F^{\prime }\left(x\right)<0\forall x\in (\frac{1}{2},3)$
$f\left(2\right)=2F\left(2\right)<2F\left(1\right)<0$, Since $F\left(x\right)$ is decreasing function in the given interval
also $f^{\prime }\left(x\right)<0\forall x\in (1,3)\Rightarrow f\left(x\right)\ne 0$ for any $xϵ(1,3)$