Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 11E / a / Solution

Choose $z_{+},z_{-}\in \mathbb{C}$ withRestrict $f$ to the line segment between them:This is continuous, with $h(0)<0<h(1)$. The intermediate value theorem gives a $t_0$ with $h(t_0)=0$. Thereforeis the required zero on a line segment.