Paragraph: Consider the function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$ 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: Which of the following is true?

$f(x)$ is decreasing on $(-1,1)$ and has a local minimum at $x=1$
$f(x)$ is increasing on $(-1,1)$ and has a local maximum at $x=1$
$f(x)$ is increasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$
$f(x)$ is decreasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$

Solution

When $x \in(-1,1)$ $ x^2 < 1 \Rightarrow x^2-1 < 0 $ $\therefore \quad f^{\prime}(x) < 0 \Rightarrow f(x)$ is decreasing. Also, at $x=1, f^{\prime \prime}(1)=\frac{4 a}{(a+2)^2>0}$ $[\because 0 < a < 2]$ $\therefore f(x)$ has a local minimum at $x=1$

Asked in: JEE Advanced 2008 (Paper 2)

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