SATHEE: Indefinite Integration 1 Question 41

Indefinite Integration 1 Question 41

42. The correct statement(s) is/are

(a) $f^{'} (1) < 0$

(b) $f (2) < 0$

(d) $f^{'} (x) = 0$ for some $x \in (1, 3)$

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Solution:

According to the given data,

$\begin{aligned} F^{'} (x) & < 0, \forall x \in (1, 3) \\ We have, & f (x) & = x F (x) \\ \Rightarrow & f^{'} (x) & = F (x) + x F^{'} (x) \\ \Rightarrow & f^{'} (1) & = F (1) + F^{'} (1) < 0 \end{aligned}$

[given $F (1) = 0$ and $F^{'} (x) < 0$ ]

$Also, f (2) = 2 F (2) < 0 [using F (x) < 0, \forall x \in (1, 3)]$

Now, $f^{'} (x) = F (x) + x F^{'} (x) < 0$

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

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Indefinite Integration 1 Question 41

42. The correct statement(s) is/are

Solution:

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