Question

# Let f(x)=⎧⎪ ⎪⎨⎪ ⎪⎩4x2+2[x]x, −12≤x<0 ax2−bx, 0≤x<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=2Given, f(x)=⎧⎪ ⎪⎨⎪ ⎪⎩4x2+2[x]x, −12≤x<0 ax2−bx, 0≤x<12 ⇒f(x)=⎧⎪ ⎪⎨⎪ ⎪⎩4x2−2x, −12≤x<0 ax2−bx, 0≤x<12 Clearly, f(x) is continuous in (−12,12)∵f(0+)=f(0−)=f(0)=0 f′(x)=⎧⎪ ⎪⎨⎪ ⎪⎩8x−2, −12≤x<0 2ax−b, 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,a∈R

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