In one variable calculus, if the derivative was not zero, it was either positive or negative and this led to increasing versus decreasing functions locally. Does this extend to complex functions and if so does this mean that there's a second derivative test for complex functions?
My intuition for this problem is that this property would extend to complex functions because we are able to find the second derivative of a complex function. The difference for complex functions is we would need to analyze the idea of "increasing" and "decreasing" in the context of multivariable calculus where we would need to find the Jacobian and apply the 2nd derivative test through that method. Is this intuition correct?