document.write( "Question 888090: For any complex number z, the minimum value of |z|+|z-1| is? \n" ); document.write( "
Algebra.Com's Answer #537215 by Edwin McCravy(20059)\"\" \"About 
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document.write( "We might suspect the minimum value of \"abs%28z%29%2Babs%28z-1%29\" is 1\r\n" );
document.write( "when z=0, as it would be if z were real.  But we have to prove it.\r\n" );
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document.write( "\"abs%28z%29%2Babs%28z-1%29\"\r\n" );
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document.write( "Let z = x+iy with x,y real.\r\n" );
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document.write( "\"sqrt%28x%5E2%2By%5E2%29%2Babs%28%28x%2Biy%29-1%29\"\r\n" );
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document.write( "\"sqrt%28x%5E2%2By%5E2%29%2Babs%28x%2Biy-1%29\"\r\n" );
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document.write( "\"sqrt%28x%5E2%2By%5E2%29%2Babs%28x-1%2Biy%29\"\r\n" );
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document.write( "\"sqrt%28x%5E2%2By%5E2%29%2Bsqrt%28%28x-1%29%5E2%2By%5E2%29\"\r\n" );
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document.write( "Clearly if this has minimum value, then y=0,\r\n" );
document.write( "since any other value of y would make it larger.\r\n" );
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document.write( "So we must find the minimum value of\r\n" );
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document.write( "\"sqrt%28x%5E2%29%2Bsqrt%28%28x-1%29%5E2%29\"\r\n" );
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document.write( "\"abs%28x%29%2Babs%28x-1%29\"\r\n" );
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document.write( "Consider the function:\r\n" );
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document.write( "\"%22f%28x%29%22\"\"%22%22=%22%22\"\"abs%28x%29%2Babs%28x-1%29\"\r\n" );
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document.write( "We consider six cases:\r\n" );
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document.write( "Case 1:  x=0    Then f(x) = f(0) = 0+1 = 1\r\n" );
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document.write( "Case 2:  x=1    Then f(x) = f(1) = 1+0 = 1\r\n" );
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document.write( "Case 3:  x>0 and x-1>0  which implies x>1\r\n" );
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document.write( "Then f(x) = x+x=1 = 2x-1\r\n" );
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document.write( "Case 4:  x>0 and x-1<0  which implies 00  which is impossible\r\n" );
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document.write( "Then f(x) = x+x=1 = 2x-1\r\n" );
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document.write( "Case 5:  x<0 and x-1>0  which is impossible\r\n" );
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document.write( "Case 6:  x<0 and x-1<0  which implies x<0\r\n" );
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document.write( "Then f(x) = -x-(x-1) = x-x+1 = -2x+1\r\n" );
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document.write( "So \"%22f%28x%29%22\"\"%22%22=%22%22\"\"abs%28x%29%2Babs%28x-1%29\" is the piecewise function:\r\n" );
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document.write( "   \"graph%28100%2C400%2F3%2C-1%2C2%2C-1%2C3%2Cabs%28x%29%2Babs%28x-1%29%29\"\r\n" );
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document.write( "Thus the minimum value is of \"abs%28z%29%2Babs%28z-1%29\" is 1.\r\n" );
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document.write( "Edwin
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