Mathematics

Question

what is the maximum vertical distance between the line y=x+2 and parabola y=x^2 for -1<=x<=2

1 Answer

  • [tex]x^2=x+2\implies x^2-x-2=(x-2)(x+1)=0\implies x=-1,x=2[/tex]

    are the intersection points of the line and parabola. [tex]x^2[/tex] is convex, which guarantees that [tex]x+2\ge x^2[/tex] over this interval. This means the vertical distance between the two functions is

    [tex]d(x)=|(x+2)-x^2|=2+x-x^2[/tex]

    Differentiating, we have

    [tex]d'(x)=1-2x[/tex]

    which has one critical point at

    [tex]1-2x=0\implies x=\dfrac12[/tex]

    At this point, we have

    [tex]d\left(\dfrac12\right)=\dfrac94[/tex]

    as the maximum vertical distance.
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