Let `f(x)={{:(,|x|-3,x lt 1),(,|x-2|+a,x ge 1):}` `g(x)={{:(,2-|x|,x lt 1),(,Sgn(x)-b,x ge 1):}` If h(x)=f(x)+g(x) is discontinuous at exactly one point, then which of the following are correct?A. a=3,b=0B. a=-3,b=-1C. a=2,b=1D. a=0,b=3

Let `f(x)={{:(,|x|-3,x lt 1),(,|x-2|+a,x ge 1):}` `g(x)={{:(,2-|x|,x l

1.	Let `f(x)={{:(,\|x\|-3,x lt 1),(,\|x-2\|+a,x ge 1):}` `g(x)={{:(,2-\|x\|,x lt 1),(,Sgn(x)-b,x ge 1):}` If h(x)=f(x)+g(x) is discontinuous at exactly one point, then which of the following are correct?A. a=3,b=0B. a=-3,b=-1C. a=2,b=1D. a=0,b=3
Answer» Correct Answer - B::C We have `f(x)={{:(,-x-3,x lt 0),(,x-3,0 le x lt 1),(,-x+2+a,1 le x lt 2),(,x-2+a,x ge 2):}` `and g(x)={{:(,2+x,x lt 0),(,2-x,0 le x lt 1),(,1-b,1 le x lt 2),(,1-b,x ge 2):}` `therefore h(x)=f(x)+g(x)={{:(,-1,x lt 0),(,-1, 0 le x lt 1),(,a-b+3-x,1 le x lt 2),(,a-b-1+x,x ge 2):}` We observe that h(x) is continuous for all the values of x, except at x=1. At x=1, it will be discontinuous, if `underset(x to 1^(-))lim h(x) ne underset(x to 1^(+))lim h(x)i.e. if-1 ne a-b+2 or, if a-bne -3` Clearly, values in options (b) and (c) satisfy this relation.