SATHEE: Application of Derivatives 4 Question 39

Application of Derivatives 4 Question 39

####41. Let $g (x) = \int_{0}^{e^{x}} \frac{f^{'} (t)}{1 + t^{2}} d t$ . Which of the following is true?

(a) $g^{'} (x)$ is positive on $(- \infty, 0)$ and negative on $(0, \infty)$

(b) $g^{'} (x)$ is negative on $(- \infty, 0)$ and positive on $(0, \infty)$

(d) $g^{'} (x)$ does not change sign $(- \infty, \infty)$

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Answer:

Correct Answer: 41. $a = \frac{1}{4}, b = \frac{- 5}{4}; f (x) = \frac{1}{4} x^{2} - \frac{5}{4} x + 2$

Solution:

$g^{'} (x) = \frac{f^{'} (e^{x})}{1 + {(e^{x})}^{2}} \cdot e^{x}$

$= 2 a \frac{e^{2 x} - 1}{{(e^{2 x} + a e^{x} + 1)}^{2}} \frac{e^{x}}{1 + e^{2 x}}$

$g^{'} (x) = 0, if e^{2 x} - 1 = 0, i.e. x = 0$

$x < 0, e^{2 x} < 1 \Rightarrow g^{'} (x) < 0$

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Application of Derivatives 4 Question 39

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