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Math Help - How do I find the slope of a tangent line that crosses two different parabolas?

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    How do I find the slope of a tangent line that crosses two different parabolas?

    Okay, so I have this one problem on my Calculus homework that I'm struggling with. I've thought about it all day long, but I'm not getting anywhere. Here's the problem word for word:

    "Two small arches have the shape of parabolas. The first is given by f(x)=1-x^2 for x= [-1, 1] and the second by g(x)= 4 - (x - 4)^2 for x=[2, 6]. A board is placed on top of these arches so it rests on both. What is the slope of the board? Hint: Find the tangent line to y=f(x) that intersects y=g(x) in exactly one point."

    I don't really want someone to answer it for me, or I'll never learn. I just need a good shove (a better shove that the "hint" in the book).
    Last edited by JonDick13926; September 26th 2013 at 05:26 PM.
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    Re: How do I find the slope of a tangent line that crosses two different parabolas?

    Have you tried the hint? How do you get the slope of the tangent lines to f(x)?
    Last edited by Prove It; September 26th 2013 at 05:25 PM.
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    Re: How do I find the slope of a tangent line that crosses two different parabolas?

    The slope of f(x) = f'(x) and the slope of g(x)=g'(x).

    And...

    f'(x)= -2x
    g'(x)= -2x+8

    That's about as far as I've been able to get all day. I've experimented with some things, like giving the points some generic names like:

    P(z,f(z))
    Q(s,g(s))

    And then setting up the change in y over change in x formula for slope and setting it equal to one of the derivatives, but that was just a shot in the dark and didn't get me far. I'm not really sure what to do.
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    Re: How do I find the slope of a tangent line that crosses two different parabolas?

     \text{Hint: If }  (x_0,y_0) \text{ is the point at which the tangent line touches } f(x), \text{ then equation of the tangent line is given by } y=(-2x_0)x+b.
    This is the starting point, and I'll check back on this later.
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