The Trapezoidal Rule for solving the differential equation

$$y^{\prime}(t)=f(t, y), \quad t \in[0, T], \quad y(0)=y_{0}$$

is defined by

$$y_{n+1}=y_{n}+\frac{1}{2} h\left[f\left(t_{n}, y_{n}\right)+f\left(t_{n+1}, y_{n+1}\right)\right]$$

where $h=t_{n+1}-t_{n}$.

Determine the minimum order of convergence $k$ of this rule for general functions $f$ that are sufficiently differentiable. Show with an explicit example that there is a function $f$ for which the local truncation error is $A h^{k+1}$ for some constant $A$.