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Homework Help: Show that y can be written as a function of x

  1. Jun 6, 2010 #1
    1. The problem statement, all variables and given/known data

    Show that y can be written as a function of x near the point (x,y)=(0,0) with

    x3+y3=6x+2y and what y'(x) is equal to.


    2. Relevant equations

    Implicit Function Theorem


    3. The attempt at a solution

    By the Implicit Function Theorem, if the partial derivative of y, Fy, does NOT equal 0 at a certain point then y can be

    written as a function of x at that certain point.

    First, I set the equation to x3+y3-6x-2y=0=F(x,y)

    Then I use the formula:

    y'(x)=(-Fx)/(Fy)

    Thus y'(x)=(-3x2+6)/(3y2-2)

    In other words, I am trying to show that the partial derivatives with respect to y

    (Fy) does NOT equal zero.

    At (0,0), Fy=3(0)2-2= -2

    And at y'(0,0)=-3


    I am not sure if I applied the Implicit Function Theorem correctly in this problem. Can

    anyone see if I miss anything? Remember I ONLY have to show that that y can be

    written as a function of x near the point (x,y)=(0,0), nothing less or more.
     
    Last edited: Jun 6, 2010
  2. jcsd
  3. Jun 7, 2010 #2
    This looks good.
     
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