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Math Help - Min. value

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    Min. value

    If Z is a Complex no. . Then Find Min. value of |2Z-1|+|3Z-2|
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    Quote Originally Posted by jacks View Post
    If Z is a Complex no. . Then Find Min. value of |2Z-1|+|3Z-2|
    Consider that |2z-1|+|3z-2|=2\cdot |z-1/2|+3\cdot |z-2/3|. Now, surely, the value of z that minimizes this expression lies on the line between z_1:= 1/2 and z_2 := 2/3 and is, therefore, a real number.
    In fact, it must be z = 2/3, for if you shift a z\in\mathbb{R} that lies between z_1 and z_2 a little closer towards z_2, without exceeding it, the first term of the sum gets larger but the second term gets smaller by a larger amount. This continues until you reach z_2; after that point on the real axis both terms increase.
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