The question is: $f:U \to \mathbb{C}$ analytic and non constant on $U$ (open and connected). Show that $|\Re(f)|$, $|\Im(f)|$ and $\Re(f)^4 + \Im(f)^4$ do not attain local maxima.
I've attempted to show this for $|\Re(f)|$ and $|\Im(f)|$ (for $f$ valued at $z \in \mathbb{C}$) via the following method:
Assume $|\Re(f)|$ attains a local maxima on $U$, then $\exists$ $z_0 \in \mathbb{C}$ such that $|\Re(f)|$ is a local maximum.
$\frac {\partial}{\partial z}|\Re(f)| = \frac {\Re(f)}{|\Re(f)|} = 0 \Rightarrow \Re(f) = 0$
So, since $f(z) = u(x,y) + iv(x,y)$ for $z=x+iy$ for $x,y\in\mathbb{R}$, then $u(x,y) = 0$ and also by $f$ being analytic we know that Cauchy-Riemann must hold $\Rightarrow v(x,y) = 0$ this suggests that $f(z) = 0 \Rightarrow$ f is constant, which is a contradiction. Thus $|\Re(f)|$ attains no local maxima (same for $|\Im(f)|$)
Would this be an acceptable way of proving this?