JEE Advanced · Mathematics · 25. AOD

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?

A
\(f(x)\) is decreasing on \((-1,1)\) and has a local minimum at \(x=1\)
B
\(f(x)\) is increasing on \((-1,1)\) and has a local maximum at \(x=1\)
C
\(f(x)\) is increasing on \((-1,1)\) but has neither a local maximum nor a local minimum at \(x=1\)
D
\(f(x)\) is decreasing on \((-1,1)\) but has neither a local maximum nor a local minimum at \(x=1\)

Verified Solution

Answer & Solution

Correct Answer

(A)
\(f(x)\) is decreasing on \((-1,1)\) and has a local minimum at \(x=1\)

Step-by-step Solution

Detailed explanation

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 \(x=1\).

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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D)	$d^{2} s p^{3}$	S)	${[F e C l_{4}]}^{2 -}$
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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?

Step-by-step Solution

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?