= Solution
Differentiating the assumed primitive identity and using <Cartan's magic formula> gives
$$
d\dot F_t=\rho_t^*\mathcal L_{X_t}\lambda
=\rho_t^*\bigl(d(\iota_{X_t}\lambda)-\iota_{X_t}\omega\bigr).
$$
Apply the <pullback of a differential form> by $\rho_t^{-1}$ and rearrange:
$$
\boxed{\iota_{X_t}\omega=dH_t,\qquad
H_t=\iota_{X_t}\lambda-\dot F_t\circ\rho_t^{-1}.}
$$
This smooth family of <Hamiltonian functions> proves that $\rho_t$ is a <Hamiltonian isotopy> in the convention of part (b).
To connect this with the preceding assertion that every $\rho_t^*\lambda-\lambda$ is an <exact differential form>, the smooth choice of primitives causes no additional obstruction. Fix $p_0\in M$ and normalize $F_t(p_0)=0$. Define $F_t(p)$ by the <line integral> of $\rho_t^*\lambda-\lambda$ from $p_0$ to $p$. The <fundamental theorem for line integrals> makes this independent of the path. Near any $p$, use one fixed path to a nearby chart center followed by coordinate line segments; this expression is smooth jointly in $t,p$. Thus these normalized primitives form the required smooth family.
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