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    Extrema and saddle points

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    Examine each function for relative extrema and saddle points:

    f (x,y) = x^2 - 3xy + y^2

    © BrainMass Inc. brainmass.com December 24, 2021, 7:46 pm ad1c9bdddf
    https://brainmass.com/math/graphs-and-functions/extrema-saddle-points-216982

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    SOLUTION This solution is FREE courtesy of BrainMass!

    We are given f(x, y) = x^2 - 3xy + y^2.

    We calculate

    f_x = 2x - 3y

    f_y = -3x + 2y

    so the critical points (a, b) are solutions to

    2a - 3b = 0

    -3a + 2b = 0

    in which case (a, b) = (0, 0) is the only solution.

    We also have

    f_xx = 2

    f_yy = 2

    f_xy = -3

    D := (f_x)(f_y) - (f_xy)^2 = (2x - 3y)(-3x + 2y) - 9

    so that D(0, 0) = - 9 < 0, in which case f does *not* have a relative extremum
    at (0, 0).

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 7:46 pm ad1c9bdddf>
    https://brainmass.com/math/graphs-and-functions/extrema-saddle-points-216982

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