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Question

If P(x1,y1), Q(x2,y2), R(x3,y3) be the points of inflection of the curve x2yx+y1=0, then number of rational vertices of the polygon PQR is

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Solution

y=x+1x2+1
dydx=(x22x+1)(x2+1)2
d2ydx2=2(x1)(x2+4x+1)(x2+1)3x=1,x=2+3,23
For these three points f′′(x) vanishes andf(x) is defined so only one rational point.

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