i) Use implicit differentiation, then set dy dx = 0.
dxd(x(x+y2))=dxdy
2x+y2+2xydxdy=dxdy
dxdy=1−2xy2x+y2
dxdy=0=>1−2xy2x+y2=0
2x+y2=0,1−2xy=0
x=−2y2 Substitute
−2y2(−2y2+y2)=y
−4y4=y y1=0,x1=0,Point(0,0)
y3=−4 y2=−34,x2=−32,Point(−32,−34)
Point(0,0),Point(−32,−34)
ii) ) The points with vertical tangents are those where the denominator of dxdy is zero (making the slope undefined). From part (i), we have
dxdy=1−2xy2x+y2=>1−2xy=0
x=2y1,y=0 Substitute
2y1(2y1+y2)=y
4y21=2y
y=234
x=232 Point(232,234)
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