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Question

If f(x)={|x|3, x<1|x2|+a, x1, g(x)={2|x|, x<2sgn(x)b, x2 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 D a=1,b=2
h(x)=f(x)+g(x)h(x)=|x|3+2|x|,x<1|x2|+a+2|x|,1x<2|x2|+a+sgn(x)b,2x<h(x)=1,x14+a2x,1x<2x1+ab,2x<

Now, for h(x) discontinuity can occur at x=1 or 2
At x=1 for h(x) to be continuous
1=4+a2a=3
At x=2 for h(x) to be continuous
4+a4=21+abb=1
h(x) will be continuous iff a=3 and b=1
For h(x) to be discontinuous at exactly one point
a=3,b1 or a3,b=1

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