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Let z be a complex number such that |z| = 1. Find the largest possible value of |z^2 + z - 1|.

Apr 16, 2019

$$|z|=1 \Rightarrow z = \cos(x)+i\sin(x)\\ z^2 + z -1 = -\sin ^2(x)+\cos ^2(x)+\cos (x)-1 +i (\sin (x)+2 \sin (x) \cos (x))\\ |z^2 + z - 1|^2 =3-2 \cos (2 x) \text{ (after much simplification) }\\ \dfrac{d}{dx} |z^2 + z - 1|^2 = 4\sin(2x)=0 \Rightarrow x = \dfrac{(2k+1)\pi}{2},~k \in \mathbb{Z}\\ \text{Thus }z = \pm i \text{ and the max value of } |z^2 + z - 1|^2 = 5 \text{ so the max value of }\\ |z^2 + z - 1| = \sqrt{5}$$