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Can someone explain this answer "in layman's terms"?

  1. Dec 30, 2014 #1
    The question is :

    Show that the curve defined implicitly by the equation

    X*Y^3 + x^3*Y = 4




    has no horizontal tangent.


    The answer is :


    4546765calcyyyyyyyyyyyyyyyyy.png


    Can someone please explain what steps are used in order to obtain the answer in a simple way for a calculus beginner? This answer is not helping me understand the question and how to answer the question ( It is too complicated and gets me confused).

    And if you have similar questions, please show me them so I can practice on them after I understand this question in order to be comfortable with this type of questions.
     
  2. jcsd
  3. Dec 30, 2014 #2
    A curve is given ##f(x,y) = c ##. To see if there are any horizontal tangents, you need to get ##y' = \frac{dy}{dx} ##. So then you take the derivative of both sides of the equation ##f(x,y) = c ## resulting in some other true equation ##f'(x,y) = 0 ##. You can obtain ##f'## by implicit differentiation. Now somewhere in ##f'## there will be a ##dy/dx## aka ##y'##. You need to extract that somehow and solve for it. How you do this will depend on the the function you were given. In this case ##y'## is that that ratio of two polynomials.

    To find if the function has a horizontal tangent we solve ##dy/dx = 0##. In our case ##dy/dx## is a fraction of two things $$y' = \frac{a(x,y)}{b(x,y)} = 0$$ can only be true if ##a(x,y)= 0##. So to find solutions set the numerator to zero (note however that ##a(x,y) = 0 ## and ##b(x,y) = 0## is a special case). Finally one goes on to see if there is a solution to ##dy/dx= y' = 0##. One finds in this case that there isn't but in general there might be.
     
    Last edited: Dec 30, 2014
  4. Jan 1, 2015 #3

    Mark44

    Staff: Mentor

    The above is not "the answer." It is the explanation of why the curve xy3 + yx3 = 4 doesn't have a horizontal tangent. The explanation seems pretty straightforward to me. What part of the explanation seems confusing to you?
     
  5. Jan 5, 2015 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Perhaps a little bit simpler: a "horizontal tangent" is where y'= dy/dx= 0. Differentiating both sides of [itex]xy^3+ x^3y= 4[/itex] with respect to x,
    [itex]y^3+ 3xy^2y'+ 3x^2y+ x^3y'= 0[/itex].

    At any point where y'= 0, that becomes [itex]y^3+ 3x^2y= y(y^2+ 3x^2)= 0[/itex]. That is only true where y= 0 or [itex]y^2+ 3x^2= 0[/itex] which, again, requires that y= 0. But if y= 0 then [itex]xy^3+ x^2y= 0[/itex], not 4. That curves does not contain any point where y= 0 so no point where y'= 0.
     
  6. Jan 5, 2015 #5

    FactChecker

    User Avatar
    Science Advisor
    Gold Member

    I think that the original explanation is pretty good and thorough, step-by-step. It uses a lot of fundamental facts. You should look at it carefully step-by-step and ask about the specific steps that are a problem.
     
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