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Math Help - Coordinates where tangent line is vertical?

  1. #1
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    Exclamation Coordinates where tangent line is vertical?

    Hey anyone know how I can solve this?

    The equation is (x^2)+2x+(y^4)+4y=5

    How do I find the coordinates of the two points on the curve where the line tangent to the curve is vertical?

    Also, is it possible for this curve to have a horizontal tangent line at the points where it intersects the x-axis? Why or why not?

    Any explanations would be great so I could learn how to approach this problem in the future. Thanks!
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  2. #2
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    Quote Originally Posted by painterchica16 View Post
    How do I find the coordinates of the two points on the curve where the line tangent to the curve is vertical?
    x^2 + 2x + y^4 + 4y = 5

    These would be the points at which x' = dx/dy = 0. Differentiating implicitly with respect to y, we have

    2xx' + 2x' + 4y^3 + 4 = 0

    (2x + 2)x' = -4y^3 - 4

    x' = \frac{-4y^3 - 4}{2x + 2}

    Setting x' = 0 yields

    \frac{-4y^3 - 4}{2x + 2} = 0 \implies -4y^3 - 4 = 0

    so y^3 = -1; y = -1.

    Substituting back into the original equation,

    x^2 + 2x + (-1)^4 + 4(-1) = 5

    x^2 + 2x + 1 - 4 = 5

    x^2 + 2x - 8 = 0

    (x + 4)(x - 2) = 0

    x = -4, 2

    So the two points are (-4, -1), and (2, -1).
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