Unable to think about a Possible application ofCR equation in complex analysis

Question

I am unable to think about an example given in my class notes related to first course in complex analysis and so I am asking for help here.

Statement of Question->It has to be shown that $\text{Re}(w)$ admits a overall maximum $z_0$ along $L$, where $L$ is $\text{Re}(z)=x_0$ .

( Above lines I am giving only for reference).

Please Note that I am only struck in how (1) below implies f(y) = $-\text{Im}(w') (x_0 +i y)$ .

Let $f(y) = \frac{d \text{Re}(w) }{dy}$ $(x_0+i y)$ --(1) and then sir wrote $f(y)= - \text{Im}(w')(x_0 +iy)$.

My attempt -> If I apply CR equation on (1)(let $f=u+iv$ then if $f$ is analytic, $u_x=v_y$ ,$v_x=- u_y$ ) using $u_y =-v_x$ . So, I am not getting $-\text{Im}(w') (x_0 +i y)$ .

Can someone please tell where I am making mistakes.

Answer 1

$$w(z)=u(z)+iv(z)\\ u(z)=\text{Re}(w);\ v(z)=\text{Im}(w)\\ f=\partial_y \text{Re}(w)_{|x_0+iy}=\partial_y u=^*-\partial_x v=-\partial_x\text{Im}(w)_{|x_0+iy}\\ $$

*: Cauchy-Riemann equations