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Question

Let f(x)=max[x2,(x1)2,2x(1x)]. If area of the region bounded by the curve y=f(x), xaxis, x=0 and x=1 is A, then the value of 54A is

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Solution

Let us draw the graph in the interval [0,1]
f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪(x1)2 ,0x132x(1x) ,13x23x2 ,23x1

Required area is,
A=1/30(x1)2 dx+2/31/32x(1x) dx+12/3x2 dx
=[(x1)33]1/30+[x22x33]2/31/3+[x33]12/3
=1981+1381+1981
=5181A=1727

54A=34

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