What is the largest absolute value attained by the function?

[Ackbach]

Here is this week's POTW:

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What is the largest absolute value attained by the function $f(z)=z^{2000}-z^4+1$ as $z$ ranges over the unit circle in the complex plane?

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[Ackbach]

Congratulations to Sudharaka for his correct solution to this week's POTW. His solution follows:

Spoiler

By the triangle inequality,

\[|z^{2000}-z^4+1|\leq |z^{2000}|+|z^4|+1\]

Since $z$ is on the unit circle, $|z|=1$. Therefore,

\[|z^{2000}-z^4+1|\leq 3\]

Notice that the maximum value of f(z) is achieved when $z=\sqrt[4]{-1}$.

\[|f(\sqrt[4]{-1})|=|(\sqrt[4]{-1})^{2000}-(\sqrt[4]{-1})^4+1|=|(-1)^{500}-(-1)+1|=3\]