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Question

If f(x)=x|x23x|e|x3|+[3+sgn (x215x)4+x2]+{13+[x22x|x|+1]}  is non differentiable at x=x1, x2, x3.....xn then ni=1xi equals
( where [.],{.},sgn(.), denotes greatest integer function, fractional part function and signum function respectively)
  1. 0
  2. 3
  3. 4
  4. 5


Solution

The correct option is B 3
sgn (x215x)=0,1,+1
3+sgn(x215x)=3,2,4
[3+sgn(x215x)4+x2]=0 and {13+[x22x|x|+1]}=13
f(x)=x|x23x|e|x3|+13
So we have to check differentiability of the function f(x) at x=0 and 3.
At x=0
For LHD
f(x)=(x33x2)e3xf(x)=(x33x2)e3x(1)+e3x(3x26x)f(0)=0
For RHD
f(x)=(x3+3x2)e3xf(x)=(x3+3x2)e3x(1)+e3x(3x2+6x)f(0)=0

At x=3
For LHD
f(x)=(x3+3x2)e3xf(x)=(x3+3x2)e3x(1)+e3x(3x2+6x)f(0)=9
For RHD
f(x)=(x33x2)ex3f(x)=(x33x2)ex3+ex3(3x26x)f(0)=9
f(x) is differentiable at x=0 but not differentiable at x=3.xi=3

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