Question

Find the coordinates of the points of the parabola that lies closest to the point

Answer 1

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Answer 2

Answer:(x,y) = (-1,1)is the closest point ony=x^2to(-3,0)Explanation:The distance from any point(x,y)to a point(hatx,haty)iscolor(white)("XXX")sqrt((x-hatx)^2+(y-haty)^2)For points ony=x^2this becomescolor(white)("XXX")d(x)=sqrt((x-hatx)^2+(x^2-haty)^2)andmore specifically for the point(hatx,haty)=(-3,0)this becomescolor(white)("XXX")sqrt((x+3)^2+(x^2-0)^2)color(white)("XX") = sqrt(x^4+x^2+6x+9)The problem is to minimized(x)or equivalently (but slightly simpler) to minimizecolor(white)("XXX")f(x)=x^4+x^2+6x+9The minimum occurs whenf'(x)= 0That is whencolor(white)("XXX")4x^3+2x+6=0An obvious (by inspection) root isx=-1(and in fact there are no other real roots)Ifx=-1theny=x^2 = (-1)^2 =1