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Math Help - Extrema points

  1. #1
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    Extrema points

    "find and classify the extreme point (x,y) of the function of two variables f(x,y) = x^2 + y^2 - xy" States the question

    My solution:

    dz/dx = 2x - y

    dz/dy = 2y - x

    y = x = 0

    therefore the extrema point is at (0,0)

    to classify it:

    D = \frac{d^2z}{dx^2} . \frac{d^2z}{dy^2} - \frac{d^2z}{dxdy}<br />
    D = 2 . 2 - (-1)^2 = 3

    3 > 0 which means the extrema is a local minima

    Am I correct?

    Thank you
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  2. #2
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    Quote Originally Posted by tsal15 View Post
    "find and classify the extreme point (x,y) of the function of two variables f(x,y) = x^2 + y^2 - xy" States the question

    My solution:

    dz/dx = 2x - y

    dz/dy = 2y - x

    y = x = 0

    therefore the extrema point is at (0,0,0)

    to classify it:

    D = \frac{d^2z}{dx^2} . \frac{d^2z}{dy^2} - \frac{d^2z}{dxdy}<br />
    D = 2 . 2 - (-1)^2 = 3

    3 > 0 which means the extrema is a local minima

    Am I correct?

    Thank you
    Yes. The graph of your function is a paraboloid with its vertex at (0, 0, 0)
    Attached Thumbnails Attached Thumbnails Extrema points-vertex_paraboloid.png  
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  3. #3
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    Quote Originally Posted by earboth View Post
    Yes. The graph of your function is a paraboloid with its vertex at (0, 0, 0)
    Why is it (0, 0, 0) ?

    Thanks for your previous and future help
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  4. #4
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    Quote Originally Posted by tsal15 View Post
    "find and classify the extreme point (x,y) of the function of two variables f(x,y) = x^2 + y^2 - xy" ...
    Quote Originally Posted by tsal15 View Post
    Why is it (0, 0, 0) ?

    Thanks for your previous and future help
    As you've stated you have a function of two (independent) variables. That means the equation

    f(x,y) = x^2 + y^2 - xy

    describes:

    f:~\begin{array}{lcl}\mathbb{R} \times \mathbb{R}&\; \mapsto \ & \mathbb{R} \\ (x,y) &\ \mapsto\ & z=f(x,y)\end{array}

    Therefore the graph of this function consists of all points (x, y, z) whose coordinates satisfy the given conditions.
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  5. #5
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    Quote Originally Posted by earboth View Post
    As you've stated you have a function of two (independent) variables. That means the equation

    f(x,y) = x^2 + y^2 - xy

    describes:

    f:~\begin{array}{lcl}\mathbb{R} \times \mathbb{R}&\; \mapsto \ & \mathbb{R} \\ (x,y) &\ \mapsto\ & z=f(x,y)\end{array}

    Therefore the graph of this function consists of all points (x, y, z) whose coordinates satisfy the given conditions.
    Of course!!! great Thank YOU
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