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Math Help - Absolute Extreme Values

  1. #1
    powerader
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    Absolute Extreme Values

    The questions reads:

    Find the absolute extreme values taken on by f(x,y) = 2x^2 -4x + y^2 -4y +3 on the triangular region bounded by the lines x = 0, y = 2, y=2x

    Steps I have done:

    gradient of f = (4x -4 , 2y -4) = 0 so then x = 1, y =2 to give the point (1,2)

    Now the three lines make up a triangle which would then have three boundaries

    B1 = { (x,y): 0 <= x <= 1, y =2 }
    parameterize to: (t,2) t is in [0,2]
    plugging back into f(t) gives me an equation, which I derive and solve for t to get the extreme value. Gives me (1,2)

    B2 = {(x,y) : 0 <= x <=1, y = 2x}
    parameterize to: (t,2t) t is in [0,2]
    plugging back into f(t) gives me an equation, which I derive and solve for t to get the extreme value. Gives me (1,2)

    How do I parameterize the last boundary which is the upward line on the y axis? I know the boundary = {(x,y): x = 0, 0 <= y <= 2 }
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  2. #2
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    Quote Originally Posted by powerader View Post
    The questions reads:

    Find the absolute extreme values taken on by f(x,y) = 2x^2 -4x + y^2 -4y +3 on the triangular region bounded by the lines x = 0, y = 2, y=2x

    Steps I have done:

    gradient of f = (4x -4 , 2y -4) = 0 so then x = 1, y =2 to give the point (1,2)

    Now the three lines make up a triangle which would then have three boundaries

    B1 = { (x,y): 0 <= x <= 1, y =2 }
    parameterize to: (t,2) t is in [0,2]
    plugging back into f(t) gives me an equation, which I derive and solve for t to get the extreme value. Gives me (1,2)

    B2 = {(x,y) : 0 <= x <=1, y = 2x}
    parameterize to: (t,2t) t is in [0,2]
    plugging back into f(t) gives me an equation, which I derive and solve for t to get the extreme value. Gives me (1,2)

    How do I parameterize the last boundary which is the upward line on the y axis? I know the boundary = {(x,y): x = 0, 0 <= y <= 2 }
    x = 0
    y = t, \, 0 \leq t \leq 2
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