Horizontal tangents via implicit differentiation

  1. 1. The problem statement, all variables and given/known data

    Find the points (if any) of of horizontal tangent lines on :

    x2 + xy + y2 = 6

    2. Relevant equations

    n/a

    3. The attempt at a solution

    So far I've concluded that I must find the points at which dy/dx = 0. I've solved for dy/dx and arrived at dy/dx = (-2x-y)/(x+2y)

    I assume that I would just have to get a "0" in the numerator to satisfy the horizontal tangent but doing so gives me

    -2x-y = 0 ==> y = -2x

    This seems that there would be an infinite number of horizontal tangents (as long as the original denominator didn't equal "0") but the graph of the original equation, per Wolfram Alpha, seems to be an ellipse so I'm only looking for two solutions...

    Have I missed a component of the concept or should I not be ending up with an ellipse?

    Thank you
     
  2. jcsd
  3. If you plug y = -2x into the original equation you don't get infinite points.
     
  4. I think I have it. In plugging in -2x for y in the original equation I get that x can be +/- sqrt(2) therefore y for x=sqrt(2) can be either -2sqrt(2) or sqrt(2) and y for x=-sqrt(2) can be either 2sqrt(2) or -sqrt(2).

    Upon substitution of all possible pairs into the derivative, I've concluded that the only two points at which dy/dx=0 are: (sqrt(2), -2sqrt(2)) and (-sqrt(2), 2sqrt(2)).

    Does this match what you have?

    Thanks again.
     
  5. HallsofIvy

    HallsofIvy 40,314
    Staff Emeritus
    Science Advisor

    Yes, that is correct.
     
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