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Math Help - Differentiability for a Function of Two or More Variables

  1. #1
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    Question Differentiability for a Function of Two or More Variables

    STATEMENT: Find all points (x,y) at which the tangent plane to the graph of z = x^2 - 6x + 2y^2 -10y + 2xy is horizontal.

    ANSWER: The answer in the back of the book is (1,2), but I have no idea how you get that answer.

    ATTEMPT: I think to start the problem you need to find the gradient vector and then use the point (0,0,1) because it's the unit vector in the z direction where it's perpendicular to the horizontal plane. That's as far as I gotten and I do not know where to go from here or if the way I started the problem is even right.

    (DEL)(which is an upside down triangle) f (x,y) = (2x - 6 +2y)i + (4y - 10 +2x)j
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Differentiability for a Function of Two or More Variables

    Solve system of the two equations:

    2x - 6 +2y=0
    4y - 10 +2x=0

    Why this will give you your answer?
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  3. #3
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    Re: Differentiability for a Function of Two or More Variables

    Thank you I got it from here.
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  4. #4
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    Re: Differentiability for a Function of Two or More Variables

    Quote Originally Posted by katchat64 View Post
    STATEMENT: Find all points (x,y) at which the tangent plane to the graph of z = x^2 - 6x + 2y^2 -10y + 2xy is horizontal.

    ANSWER: The answer in the back of the book is (1,2), but I have no idea how you get that answer.

    ATTEMPT: I think to start the problem you need to find the gradient vector and then use the point (0,0,1) because it's the unit vector in the z direction where it's perpendicular to the horizontal plane. That's as far as I gotten and I do not know where to go from here or if the way I started the problem is even right.

    (DEL)(which is an upside down triangle) f (x,y) = (2x - 6 +2y)i + (4y - 10 +2x)j
    The LaTeX command for "del" is \nabla: \nabla f(x,y)= (2x- 6+ 2y)i+ (4y- 10+ 2x)j

    But what I would do is define F(x,y,z)=  x^2 - 6x + 2y^2 -10y + 2xy- z so that \nabla F(x, y, z)= (2x- 6+ 2y)i+ (4y- 10+ 2x)j- k and then argue that the tangent plane will be parallel to the xy-plane if and only if that vector is in the z-direction. That is, that 2x- 6+ 2y= 0 and 4y- 10+ 2x= 0 as Also sprach Zarathustra said.
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