If f(x)={|x|−3,x<1|x−2|+a,x≥1,g(x)={2−|x|,x<2sgn(x)−b,x≥2 and h(x)=f(x)+g(x) is discontinuous at exactly one point, then which of the following values of a and b are possible
A
a=−3,b=0
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B
a=2,b=1
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C
a=−3,b=2
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D
a=1,b=2
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Solution
The correct option is Da=1,b=2 h(x)=f(x)+g(x)⇒h(x)=⎧⎨⎩|x|−3+2−|x|,x<1|x−2|+a+2−|x|,1≤x<2|x−2|+a+sgn(x)−b,2≤x<∞⇒h(x)=⎧⎨⎩−1,x≤14+a−2x,1≤x<2x−1+a−b,2≤x<∞
Now, for h(x) discontinuity can occur at x=1 or 2
At x=1 for h(x) to be continuous ⇒−1=4+a−2⇒a=−3
At x=2 for h(x) to be continuous ⇒4+a−4=2−1+a−b⇒b=1 h(x) will be continuous iff a=−3 and b=1 ∴ For h(x) to be discontinuous at exactly one point a=−3,b≠1 or a≠−3,b=1