Maximum/mimimum of a complex function

[mvillagra]

Hi, I have the following problem

given a function f(k) defined on the reals and a complex constant z0, what is the maximum of the following function?

$$z_0f(k)$$

The maximum of the module is clearly the value k such that

$$z_0f'(k)=0$$

right? because when you take the module, the squares of the real and imaginary parts are maximum and hence the module is maximum.

But what happens when you cannot factorize the complex constants? e.g., given the following fuction

$$g(k)=\sqrt{z_1+z_2\sin k}$$

where k is real, and z1 and z2 are complex constants. Can we still derivate g, make it equal to 0 and still say we can find a critical point? i.e., does solving
$$g'(k)=0$$
gives you a critical point?

thanks in advance for the help :shy:

 

[foxjwill]

Well, first of all, you seem to want the maximum of the absolute value. In this case, you'll need to find the critical points not of $$z_0f(k)$$ (or for the second case, g(k)), but of $$|z_0f(k)|$$ (or for the second case, |g(k)|).

 

[mvillagra]

I didn't write it down correctly, but the question is

does solving |g(k)|'=0 and g'(k)=0 give the same critical points?

 

[foxjwill]

Have you tried finding the critical points of each and comparing them?