A different approach to the question: Determine max $|z^3 − z + 2|$. $z∈\Bbb C, |z|=1 $

Asked Jun 28 '21 at 04:05

Active Jun 28 '21 at 08:33

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Determine max $|z^3 − z + 2|$.
$z∈\Bbb C, |z|=1 $

Now, quite clearly, substituting $z = x + iy$ makes the question easy. The solution is not very difficult after that. However, is there some other (shorter or longer) method to solve it and is a geometrical perspective possible?

Finally, have you seen something similar being used practically? These questions to maximize an expression are quite common.

edited Jun 28 '21 at 08:33

asked Jun 28 '21 at 04:05

mathx

1

See the second answer here for a trig solution. – dxiv Jun 28 '21 at 04:54
@dxiv Useful... – mathx Jun 28 '21 at 05:08
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Does this answer your question? How find $\max _{z: \ |z|=1} \ f \left( z \right)$ for $f \left( z \right) = |z^3 - z +2|$ – Martin R Jun 28 '21 at 05:17
@MartinR Thanks for that. Do you have something to mention regarding the other parts of the question? – mathx Jun 28 '21 at 05:25
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You can interpret complex numbers geometrically ($z$ is on the complex unit circle, $1/z$ is reflection in the x axis, magnitude is distance to origin, etc.) However, it’s rare for algebra to have a nice geometric interpretation and more common for complex numbers/algebra being used to bash geometric problems. Maximization problems are common, but solving them is usually easily done with calculus or numerical approximation. – Eric Jun 28 '21 at 12:27
@Eric Thanks for that. – mathx Jun 28 '21 at 12:43

A different approach to the question: Determine max $|z^3 − z + 2|$. $z∈\Bbb C, |z|=1 $

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