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Question

Let f(x) be the function defined by f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪limn(1+[sinxx]2n){x}1+[xtanx]2n;0<x<π2{x}33{x}4[x];2<x0x327x46x2⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
(where [.] denotes greatest integer function and {.} denotes fractional part function) then which of the following options is/are true?
  1. There exist exactly 2 points of local minima of f(x) in the interval (4,π2).
  2. The sum of abscissa of all of its point of local maxima is -4
  3. The sum of abscissa of all of its point of local minima is 1
  4. f(x) is not differentiable in the interval (,1)


Solution

The correct options are
B The sum of abscissa of all of its point of local maxima is -4
C The sum of abscissa of all of its point of local minima is 1
D f(x) is not differentiable in the interval (,1)
f(x)=⎪ ⎪ ⎪⎪ ⎪ ⎪{x};0<x<π2{x}33{x}4[x];2<x0x327x46;x2⎪ ⎪ ⎪⎪ ⎪ ⎪f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪x327x46,x2(x+2)3(x+2)6,2<x<1(x+1)3(x+1)5,1x<00;x=0x;0<x<1x1;1x<π2⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

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