Let z be a complex number with |z|=1. Find the maximum value of |z^3-z+2|.

Write $z=e^{i\theta}$ and expand the square of the modulus: $|z^3-z|^2=2-2\cos2\theta$ and the cross term is $2\,\mathrm{Re}\big((z^3-z)\cdot2\big)=4(\cos3\theta-\cos\theta)$, so $|z^3-z+2|^2=6-2\cos2\theta+4\cos3\theta-4\cos\theta$. With $c=\cos\theta\in[-1,1]$ this becomes $g(c)=16c^3-4c^2-16c+8$. Then $g'(c)=48c^2-8c-16=0$ gives $6c^2-c-2=0$, so $c=\frac23$ or $c=-\frac12$. Evaluating, $g(-\tfrac12)=-2-1+8+8=13$, $g(\tfrac23)=\tfrac{8}{27}\approx0.30$, $g(1)=4$, $g(-1)=4$. The maximum of $g$ is $13$, so the maximum modulus is $\sqrt{13}\approx3.6056$, i.e. $3.61$. A direct scan over $\theta$ confirms this. The endpoint values $g(1)$ and $g(-1)$ are smaller than $13$, and the critical point $c=\tfrac23$ gives a local minimum of $g$, so the global maximum on the interval is indeed at $c=-\tfrac12$. This corresponds to $z=e^{2\pi i/3}$ or its conjugate.