what is the maximum vertical distance between the line y=x+2 and parabola y=x^2 for 1<=x<=2
Mathematics
sandipkc841
Question
what is the maximum vertical distance between the line y=x+2 and parabola y=x^2 for 1<=x<=2
1 Answer

1. User Answers LammettHash
[tex]x^2=x+2\implies x^2x2=(x2)(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+xx^2[/tex]
Differentiating, we have
[tex]d'(x)=12x[/tex]
which has one critical point at
[tex]12x=0\implies x=\dfrac12[/tex]
At this point, we have
[tex]d\left(\dfrac12\right)=\dfrac94[/tex]
as the maximum vertical distance.