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Question

Let f(x)=sinπxx2, x>0.
Let x1<x2<x3<...<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?

A
x1<y1
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B
xn+1xn>2 for every n
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C
xn(2n,2n+12) for every n
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D
|xnyn|>1 for every n
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Solution

The correct option is D |xnyn|>1 for every n
f(x)=sinπxx2
f(x)=πx2cosπx2xsinπxx4=2cosπx(πx2tanπx)x3 where x0
For maxima and minima: f(x)=0
cosπx(πx2tanπx)=0
Let g(x)=cosπx or, h(x)=tanπxπx2
From the above graph we can observe that product of both g(x) and h(x) changes its sign at given interval
πx=(2n+1)π2
x=2n+12, nI
From the graph, we can see that for all x=2n+12, doesn't change sign. Also we can see x1 is point of maxima and y1 is point of minima and x1>y1
Same is the case for all x satisfying πx2=tanπx
where yn(2n1,2n12) n=1,2,3....
and xn(2n,2n+12) n=1,2,3....
|xnyn|>1 for every n
x1>y1
xn+1yn+1>1 and yn+1xn>1
xn+1xn>2

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