Differentiation and its Application: Which of the following is true? | JEE Advanced 2008 Paper 2

Comprehension · Questions 1–1

Read the passage given below and answer the questions that follow.

Consider the function $f:(-\infty,\infty)\to(-\infty,\infty)$ defined by $$f(x)=\frac{x^2-ax+1}{x^2+ax+1},\quad 0<a<2.$$

JEE Advanced 2008 Paper 2Question 1

Which of the following is true?

Differentiation and its Application

$f(x)$ is decreasing on $(-1,1)$ and has a local minimum at $x=1$✓ Correct answer$f(x)$ 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

Concept. Here $f(x)=\dfrac{x^2-ax+1}{x^2+ax+1}$ with $0<a<2$, and $f'(x)=\dfrac{2a(x^2-1)}{(x^2+ax+1)^2}$. The sign of $f'$ is the sign of $x^2-1$. Formulas used. (1) $f'$ changes $-\to+$ at a local minimumStep 1 - On $(-1,1)$. $a>0$ and $x^2-1<0$, so $f'<0$: $f$ is decreasing. Step 2 - At $x=1$. $f'<0$ just before $1$ and $f'>0$ just after. By (1), a local minimum. Answer: (A)

Differentiation and its Application - JEE Mathematics question (JEE Advanced 2008 Paper 2)

More from Differentiation and its Application