Maximum value of complex valued function

Question

I need to find maximum value of the following complex valued function \begin{align*} \underset{-\pi\le x\le \pi}{\mathrm{max}}\left\vert x e^{-\mathrm{i}k x^2}\right\vert,\quad k\in \mathbb{R}^{+}. \end{align*} I do not know how to proceed.

Answer 1

We know that

$$\forall t\in \Bbb R\;\;\; e^{it}=\cos (t)+i\sin (t) $$ and

$$|e^{it}|=\sqrt {\cos^2 (t)+\sin^2 (t)}=1$$

thus $$|xe^{-ikx^2}|=|x| |e^{-ikx^2}|=|x|$$

the max is $\pi $.