My first problem on complex numbers

Algebra Level 5

Let z z be a complex number with ∣ z ∣ = 2 \left| z \right| =\sqrt { 2 } then find the maximum value of ∣ ( z − 1 ) 2 ( z + 1 ) z 2 ∣ \Large\left| \frac { (z-1)^{ 2 }(z+1) }{ { z }^{ 2 } } \right| .

Write your answer as the square of the number.


The answer is 8.

Hamza A by Hamza A , 19, USA

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2 solutions

Otto Bretscher
Mar 20, 2016

I took a straightforward approach: If we let z = x + i y z=x+iy and work it out, the square of the expression comes out to be ( 3 − 2 x ) 2 ( 2 x + 3 ) 4 \frac{(3-2x)^2(2x+3)}{4} , with a maximum of 8 \boxed{8} at x = − 1 2 x=-\frac{1}{2} in the domain ∣ x ∣ ≤ 2 |x|\leq \sqrt{2}

8 Helpful 0 Interesting 0 Brilliant 1 Confused
Subh Mandal
Apr 5, 2020

more general approach, substitute z = re^ix and differentiate the magnitude function wrt x. to get 8 trivially

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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