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Question

Let f(x) is a continuous function which takes positive values for x (x>0), and satisfy x0f(t)dt=xf(x) with f(1)=12. Then the value of f(2+1) equals


A
1
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B
21
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C
14
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D
121
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Solution

The correct option is C 14
We have x0f(t)dt=xf(x) ....(1)
Differentiating both the sides of equation (1) w.r.t. x, we get
f(x)=xfx2f(x)+f(x);Letf(x)=y2f(x)=2ydydxy2=x.2y.dydx.12y+yy2=x.dydx+yy2y=x.dydxdyy(y1)=dxxy(y1)y(y1)dy=dxx;ln(y1)y=lncx(y1)y=cx
11y=cx1y=1cxy=11cxf(x)=11cx ....(1)
If x=1,f(1)=12 (given)
1c=2;c=12f(x)=11+(21)xf(x)=1[1+(21)x]2f(2+1)=14

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