  Question

# Let f(x) be the function defined by f(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩limn→∞(1+[sinxx]2n){x}1+[xtanx]2n;0<x<π2{x}3−3{x}4−[x];−2<x≤0x3−27x−46x≤−2⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭ (where [.] denotes greatest integer function and {.} denotes fractional part function) then which of the following options is/are true?There exist exactly 2 points of local minima of f(x) in the interval (−4,π2).The sum of abscissa of all of its point of local maxima is -4The sum of abscissa of all of its point of local minima is 1f(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}3−3{x}4−[x];−2<x≤0x3−27x−46;x≤−2⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭⇒f(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩x3−27x−46,x≤−2(x+2)3−(x+2)6,−2<x<−1(x+1)3−(x+1)5,−1≤x<00;x=0x;0<x<1x−1;1≤x<π2⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭  Suggest corrections   