Indefinite Integration 1 Question 41

42. 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:

  1. 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 $F(x)<0, \forall x \in(1,3)$ ]



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