<s>[INST] How find $\max _{z: \ |z|=1} \ f \left( z \right)$ for $f \left( z \right) = |z^3 - z +2|$ Let $f : C \mapsto R $,  $f \left( z \right) = |z^3 - z +2|$. How find $\max _{z: \ |z|=1} \ f \left( z \right)$ ? 
 [/INST] $\bf{My\; Solution::}$ Let $z=x+iy\;,$ Then $|x+iy| = 1\Rightarrow x^2+y^2 =1 $
Now We have To Maximize $$f(z) = \left|z^3-z+2\right| = \left|(x+iy)^3-(x+iy)+2\right|$$
We Get
$$f(x,y) = \left|x^3-iy^3+3ix^2y-3xy^2-x-iy+2\right|$$
$$f(x,y)=\left|x(x^2-3y^2)+iy(3x^2-y^2)-x-iy+2\right|\;,$$ Using $x^2+y^2 = 1$
$$f(x,y) = \left|x(4x^2-3)+iy(4x^2-1)-x-iy+2\right|=\sqrt{[(4x^3-4x)-2]^2+(4x^2y)^2}$$
Now Let 
$$g(x,y) = (4x^3-4x-2)^2+4x^4y^2 = (4x^3-4x-2)^2+4x^4(1-x^2)\;,$$ Using $x^2+y^2 = 1$
So Here we have to Maximize $$g(x) = (4x^3-4x-2)^2+4x^4-4x^6$$
Using Derivative Test, You Can Maximize It.
</s>