IIT-JEE 2008 · Paper 2 · Mathematics Q15

MathematicsSingle correct+4 / -1Question 15 of 66

Passage

Paragraph for Question Nos. 14 to 16

Consider the function $f:(-\infty,\infty)\to(-\infty,\infty)$ defined by
$f(x)=\frac{x^2-ax+1}{x^2+ax+1},\quad0<a<2.$

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

Answer and worked solution

Official answer: A

Sign of $f'$ . $f'(x)=\dfrac{2a(x^2-1)}{D^2}$ (with $D>0$ since $0<a<2$ ) is negative on $(-1,1)$ and positive for $|x|>1$ .

So $f$ decreases on $(-1,1)$ and then increases: a local minimum at $x=1$ .

(A)

Topics:Monotonicity Maxima and minima

IIT-JEE 2008 · Paper 2, Mathematics question 15