[tex][|z - 3|^2 + |z - 5 + 2i|^2 + |z - 1 + i|^2][/tex]
Let , complex number z be , z = x + iy .
Putting z in above equation , we get :
[tex]=(x-3)^2+y^2+(x-5)^2+(y+2)^2+(x-1)^2+(y+1)^2[/tex]
Now , getting critical points by :
[tex]f'(x) = 2(x-3)+2(x-5)+2(x-1) = 0\\\\3x-9=0\\\\x=3[/tex]
Also ,
[tex]f'(y)=2y+2(y+2)+2(y+1)=0\\\\3y+3=0\\\\y=-1[/tex]
So , at point ( 3, -1 ) complex number given expression have minimum value.
[tex]=(3-3)^2+(-1)^2+(3-5)^2+(-1+2)^2+(3-1)^2+(-1+1)^2\\\\=10[/tex]
Therefore, minimum value is 10.
Hence, this is the required solution.
