Let f(x) = (sin π x/x2), x 0. Let x1

Question

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? (A) x1 < y1 (B) xn+1 - xn > 2 for every n (C) xn ∈ (2n, 2n +(1/2)) for every n (D) |xn-yn | > 1 for every n

Tardigrade Admin · Accepted Answer

f (x)=(sin π x/x2) f '(x)=(2x cos π x((π x/2)-tan π x)/x4) ⇒ |xn-yn| > 1 for every n x1 > y1 xn ∈ (2n, 2n + 12) xn+1 -xn > 2.

Q. Let
$f (x) = \frac{s in π x}{x ^{2}}, x > 0.$
Let $x_{1} < x_{2} < 3 < \dots < x_{n} < \dots$ be all the points of local maximum of f
and $y_{1} < y_{2} < y_{3} < \dots < y n < \dots$ be all the points of local minimum of f.
Then which of the following options is/are correct?

$x_{1} < y_{1}$

$x_{n + 1} - x_{n} > 2$ for every n

$x_{n} \in (2 n, 2 n + \frac{1}{2})$ for every n

$∣ x_{n} - y_{n} ∣ > 1$ for every n

$f (x) = \frac{s in π x}{x ^{2}}$
$f^{'} (x) = \frac{2 x cos π x ( \frac{π x}{2} - t an π x )}{x ^{4}}$
$\Rightarrow ∣ x_{n} - y_{n} ∣ > 1$ for every $n$
$x_{1} > y_{1}$
$x_{n} \in (2 n, 2 n + 12)$
$x_{n + 1} - x_{n} > 2.$

Q. Let f(x)=x2sinπx​,x>0. Let x1​<x2​<3<⋯<xn​<⋯ be all the points of local maximum of f and y1​<y2​<y3​<⋯<yn<⋯ be all the points of local minimum of f. Then which of the following options is/are correct?

x1​<y1​

xn+1​−xn​>2 for every n

xn​∈(2n,2n+21​) for every n

∣xn​−yn​∣>1 for every n