Find Max Of $M=|z^{3}-z+2|$ Give a complex number $z$: $|z|=1$
Find Max $M=|z^{3}-z+2|$
Could someone help me solve this ?

Let $z=x+yi$, hence $$\displaystyle \small \left|z^3-z+2\right|=\left|(x+yi)^3-(x+yi)+2\right|=\left|x^3+3x^2yi-3xy^2-y^3i-x-yi+2\right|= \\ \small =\left|(x^3-3xy^2-x+2)+i(3x^2y-y^3-y)\right|=\left|(x^3-3xy^2-x+2)+iy(3x^2-y^2-1)\right|$$
It is given that $|z|=1 \ \mathrm{thus\,} \ x^2+y^2=1 \rightarrow y^2=1-x^2$.
Setting it to the equation we get $$\displaystyle \small M=\left|z^3-z+2\right|=\left|(x^3-3x(1-x^2)-x+2)+iy(4x^2-2)\right|=\left|(4x^3-4x+2)+iy(4x^2-2)\right|$$
Now we can evaluate and get $$\displaystyle \small M=\sqrt{(4x^3-4x+2)^2+y^2(4x^2-2)^2}=\sqrt{(4x^3-4x+2)^2+(1-x^2)(4x^2-2)^2}$$
Can you proceed?
