Definite Integration Question 41
Question 41
- 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 $x \in(1,3)$
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Solution:
- According to the given data,
$$ \begin{aligned} & & F^{\prime}(x) & <0, \forall x \in(1,3) \ \text { We have, } & & f(x) & =x F(x) \ \Rightarrow & & f^{\prime}(x) & =F(x)+x F^{\prime}(x) \ \Rightarrow & & f^{\prime}(1) & =F(1)+F^{\prime}(1)<0 \end{aligned} $$
[given $F(1)=0$ and $F^{\prime}(x)<0$ ]
$$ \text { Also, } \quad f(2)=2 F(2)<0 \quad \text { [using } F(x)<0, \forall x \in(1,3)] $$
Now, $f^{\prime}(x)=F(x)+x F^{\prime}(x)<0$
[using $F(x)<0, \forall x \in(1,3)$ ]
