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Question

Let f(x)=⎪ ⎪⎪ ⎪4x2+2[x]x, 12x<0 ax2bx, 0x<12 where [x] denotes the greatest integer function. Then which of the following options is correct

A
f(x) is continuous and differentiable in (12,12) for all real a, provided b=2
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B
f(x) is continuous and differentiable in (12,12) for all real b, provided a=2
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C
f(x) is continuous and differentiable in (12,12) iff a=4 , b=2
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D
For no real value of a and b,f(x) is differentiable in(12,12)
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Solution

The correct option is A f(x) is continuous and differentiable in (12,12) for all real a, provided b=2
Given, f(x)=⎪ ⎪⎪ ⎪4x2+2[x]x, 12x<0 ax2bx, 0x<12
f(x)=⎪ ⎪⎪ ⎪4x22x, 12x<0 ax2bx, 0x<12

Clearly, f(x) is continuous in (12,12)f(0+)=f(0)=f(0)=0

f(x)=⎪ ⎪⎪ ⎪8x2, 12x<0 2axb, 0<x<12
For f(x) to be differentiable , At x=0,L.H.D must be equal to R.H.D.
8(0)2=2(a)(0)b
b=2,aR

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