1. Tardigrade
  2. Question
  3. Mathematics
  4. Let f(x) = (sin π x/x2), x > 0. Let x1< x2 < 3 <⋅s < xn < ⋅s be all the points of local maximum of f and y1 < y2 < y3 < ⋅s < yn < ⋅s be all the points of local minimum of f. Then which of the following options is/are correct?

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Q. Let
$f\left(x\right) = \frac{sin\,\pi x}{x^{2}},\quad x > 0.$
Let $x_{1}< x_{2} < 3 <\cdots < x_{n} < \cdots$ be all the points of local maximum of f
and $y_{1} < y_{2} < y_{3}^{ }< \cdots < yn < \cdots$ be all the points of local minimum of f.
Then which of the following options is/are correct?

JEE AdvancedJEE Advanced 2019Application of Derivatives

Solution:

$f \left(x\right)=\frac{sin\,\pi x}{x^{2}}$
$f '\left(x\right)=\frac{2x\,cos\,\pi x\left(\frac{\pi x}{2}-tan\,\pi x\right)}{x^{4}}$
$\Rightarrow \left|x_{n}-y_{n}\right| > 1$ for every $n$
$x_{1} > y_{1}$
$x_{n}\,\in\,\left(2n, 2n + 12\right)$
$x_{n+1} -x_{n} > 2.$

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