Paragraph: Consider the function f:(-∞, ∞) arrow(-∞, ∞) defined by f(x)=(x^2-a x+1)/(x^2+a x+1) ; 0 Question: Let g(x)=∫_0^ e^x f^ (t) 1+t^2 d t . Which of the following is true ?

Correct answer: B. g^ (x) is negative on (-∞, 0) and positive on (0, ∞) g^ (x)= f^ (e^x) 1+(e^x)^2 · e^x=2 a[ e^ 2 x -1 (e^ 2 x +a e^x+1)^2 ]( e^x 1+e^ 2 x ) aligned & g^ (x)=0 , if e^ 2 x -1 & =0, i.e., & & x=0 If & & x < 0, e^ 2 x < 1 g^ (x) < 0 . aligned

JEE Advanced 2008Paper 2

Exam: JEE Advanced
Year: 2008
Paper: Paper 2
Subject: Mathematics
Chapter: Application of Derivatives

JEE Advanced 2008 · Paper 2 · Mathematics · Application of Derivatives

Paragraph: Consider the function f:(-∞, ∞) arrow(-∞, ∞) defined by...

Paragraph: Consider the function

f:(-\infty, \infty) \rightarrow(-\infty, \infty)

defined by

f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 < a < 2

Question: Let

g(x)=\int_0^{e^x} \frac{f^{\prime}(t)}{1+t^2} d t

. Which of the following is true ?

Options

A $g^{\prime}(x)$ is positive on $(-\infty, 0)$ and negative on $(0, \infty)$
B $g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$
C $g^{\prime}(x)$ changes sign on both $(-\infty, 0)$ and $(0, \infty)$
D $g^{\prime}(x)$ does not change sign on $(-\infty, \infty)$

Correct answer

B. $g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$

Step-by-step solution

\because g^{\prime}(x)=\frac{f^{\prime}\left(e^x\right)}{1+\left(e^x\right)^2} \cdot e^x=2 a\left[\frac{e^{2 x}-1}{\left(e^{2 x}+a e^x+1\right)^2}\right]\left(\frac{e^x}{1+e^{2 x}}\right)

\begin{aligned} & g^{\prime}(x)=0 \text{, if } e^{2 x}-1 & =0, \\ \text{ i.e., } & & x=0 \\ \text{ If } & & x < 0, e^{2 x} < 1 \Rightarrow g^{\prime}(x) < 0 .\end{aligned}

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