Calc I derivatives, I know the answers, but how did I get there?

[i2c]

Homework Statement

Question #2
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Homework Equations

The Attempt at a Solution

I can of course find the derivatives for both functions. When I set them equal to each other I get x=1/2 that's where the functions have parallel tangent lines for the same x value, however (I think that) the functions have parallel tangents everywhere, just at different x values. So I figured out that at any point x on x^2, there will be a parallel tangent from -x^2+2x-5 at 1-x

I guess you have to find the equation of a tangent line that intersects with both graphs only once, no more no less, but I don't know how to write that out and solve for it. The answers are the tangent lines when x = -1 and 2 on x^2

 

[vela]

Label the point where the line intersects the upper parabola (a, a²) and the point where it intersects the lower parabola (b, -b²+2b-5). So those are two points on the line, and you also found that b=1-a. That should be enough to figure out the equation of the line, I think.