Let $f$ be a real valued differentiable function of $(0,1).$ Set $g=f'+if$, where $f'$ denotes derivatives of $f$ and $i^2=-1.$ Let $a,b\in(0,1)$ be two consecutive zeros of $f$. Which of the following statements is/are true.
$(1).$ if $g(a)>0$, then $g$ crosses the real line from upper half plane to lower half plane at $a.$
$(2).$ if $g(a)>0$, then $g$ crosses the real line from lower half plane to upper half plane at $a.$
$(3).$ if $g(a)g(b)\neq 0$, then $g(a), g(b)$ have same sign.
$(4).$ if $g(a)g(b)\neq 0$, then $g(a), g(b)$ have opposite sign.
I am trying it by taking $f(x)=(x-a)(x-b)=x^2-(a+b)x+ab$, then $g=2x-(a+b)+i(x^2+(a+b)x+ab).$ Now condition $g(a)>0$ says that $a>b.$ How this conditions will says that option $2$ is correct. In answer key options $2$ and $4$ are given correct. Please help. Thanks.