Find the parabola of the form y=ax^2 +bx that shares a tangent line with the function y= (2x)/(x^2 +1) when x=1. Solve and show all steps.

Question

Find the parabola of the form y=ax^2 +bx that shares a tangent line with the function y= (2x)/(x^2 +1) when x=1. Solve and show all steps.

Answer 1

First, the tangent line.

y = (2x)/(x^2+1)
y' = -2(x^2-1)/(x^2+1)^2
At x=1, y=1 and y'=0
So, the tangent line at x=1 is the line y=1

Now, surely you can come up with a parabola which has its vertex at (1,1) and goes through the origin.