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Question

Let f:RR be a function defined as

f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪sin(a+1)x+sin2x2x,if x<0 b,if x=0x+bx3xbx5/2,if x>0

If f is continuous at x=0, then the value of a+b is equal

A
2
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B
52
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C
32
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D
3
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Solution

The correct option is C 32
Given : f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪sin(a+1)x+sin2x2x,if x<0 b,if x=0x+bx3xbx5/2,if x>0

f is continuous at x=0, then
f(0)=f(0)=f(0+)
Now,
f(0)=limx0sin(a+1)x+sin2x2xf(0)=a+12+1=a+32 (1)

f(0+)=limx0+x+bx3xbx5/2f(0+)=limx0+bx3bx5/2(x+bx3+x)f(0+)=limx0+bb(1+bx2+1)f(0+)=12(2)
From equations (1) and (2), we get
a=2, b=12a+b=32

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