Find and classify all the extreme points for the function f(x,y) =𝑥^2 +𝑦^2 -xy+x.?

2 Answers

  • Pope
    Lv 7
    4 weeks ago
    Favourite answer

    Level curves are in the form x² - xy + y² + x = k, which is an ellipse if it is anything.

    That makes the surface z = f(x, y) an elliptic paraboloid. It must have one extremum.

    Intersecting it with vertical plane x = 0 results in parabola z = y², which has no maximum. The single extremum must be a minimum.

    Find the partial derivatives, and let each of them equal zero.

    ∂/∂x f(x, y) = 2x - y + 1

    ∂/∂y f(x, y) = -x + 2y

    Solve this system:

    2x - y + 1 = 0

    -x + 2y = 0

    x = -2/3

    y = -1/3

    f(-2/3, -1/3) = -1/3

    Function f has one minimum, -1/3. It has no other extrema.

  • rotchm
    Lv 7
    4 weeks ago

    State here the procedure you saw for such problems. Then we will proceed.

    Hopefully no one will spoil you the answer thereby depriving you from your personal enhancement; that would be very inconsiderate of them. 

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