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Question

Let f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪x2a;0x<1a;1x<22b24bx2;2x<
If f(x) is continuous for 0x<, then the most suitable values of a and b are

A
a=1,b=1
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B
a=1,b=1+2
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C
a=1,b=1
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D
none of these
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Solution

The correct option is D a=1,b=1
Given f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪x2a0x<1a,1x<22b24bx2,2x<
Since, f(x) is continuous in 0x<. So f(x) is continuous at x=1
f(1)=LHL=RHL
Here, f(1)=a
LHL=limx1f(x)=limh0f(1h)
=limh0(1h)2a=1a
1a=aa2=1
a=±1
Also, f(x) is continuous at x=2
LHL=limx2f(x)
LHL=a=±1
f(2)=2b24b2=b22b
For a=1
1=b22b
b22b+1=0
b=1
For a=1
b22b1=0
(b1)2=2
b=1±2

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