Unformatted text preview: / 4 + 1 / 2 = 5 / 4, attained at the point (1 , 1 / 2). On side 3, 0 ≤ x ≤ 1, y = 1, and u ( x, y ) = u ( x, 1) = x 2 + x1 = f ( x ). On this side, f ± ( x ) = 2 x + 1, f ± ( x ) = 0 ⇒ x =1 / 2. Extremum values occur at the endpoints: f (0) =1, f (1) = 1. Thus the minimum value is1 and is attained at the point (0 , 1). Maximum value is 1 and is attained at the point (1 , 1). On side 4, 0 ≤ y ≤ 1, x = 0, and u ( x, y ) = u (0 , y ) =y 2 . On this side, the maximum value is 0 and is attained at the point (0 , 0), and the minimum value is1 and is attained at the point (0 , 1). Consequently, the maximum value of u on the square is 5 / 4 and is attained at the point (1 , 1 / 2); and the minimum value of u on the square is1 and is attained at the point (0 , 1) (see Fgure). Plot3D x^2 y^2 x y, x, 0, 1 , y, 0, 1 , ViewPoint 2, 2, 1 1...
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 Fall '11
 StuartChalk
 maximum value, minimum value, Conformal mappings, uyy

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