Rolle's Theorem and error bound formula for trapezoidal rule

Question

In these notes on error bounds for numerical integration, it shows proofs for the error bound formula for the midpoint rule and the trapezoidal rule.

I don't understand the last part (page 5-6) for the trapezoidal rule.

1. What's the point of using Rolle's theorem?

2. Why does he suddenly use a $G(x)$ function with that "$q(...)$" ?

3. More specifically, what's the goal of using $G''(x) = 0\,$?

https://www.macmillanlearning.com/studentresources/highschool/mathematics/rogawskiapet2e/additional_proofs/error_bounds_proof_for_numerical_integration.pdf