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What is the minimum value of |2z-1| + | 3z -2| ? I have been trying it for days but am not able to get the answer. Please help.

What is the minimum value of |2z-1| + | 3z -2| ? I have been trying it for days but am not able to get the answer. Please help.

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Grade:11

1 Answers

Riddhish Bhalodia
askIITians Faculty 434 Points
8 years ago
Ok this is a slightly difficult to explain but I will try

Part 1: This is straight forward
2|z-1/2| + 3|z-2/3|
now considering these two terms are positive we can apply AM GM and the equality exist when terms are equal
2|z-1/2| = 3|z-2/3|

Part 2: Now we need to solve this, thinking intutively the smallest such z would lie between the points (1/2,0) and (2/3,0) cutting it in ratio 3:2
that point can be found easily to be
(3/5,0)
Hence2|z-1/2| = 3|z-2/3| = 1/5
Thus the minimum value is 2/5

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