If `f (x)= {{:(|x|-3"," , x lt 1 ), (|x-2| + a"," , x ge 1):},g (x) = {{:(2-|x|"," , x lt 2),( sgn (x)-b"," , x ge 2):}If h (x) = f(x)+ g (x)` is discontinous at exactly one point, then which of the following are correct ?

Options

A. a=3,b=0
B. a=-3,b=-1
C. a=2,b=1
D. a=0,b=3

Answer

B, C

Explanation

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.