Looking for a partial converse of Rolle's theorem

Question

Let $f: [a,b] \to \mathbb R $ be a continuous function differentiable in $(a,b)$ such that $f(b)=0$ and for some $c \in (a,b) , f'(c)=0$ ; then under what additional conditions can we conclude that $\exists d \in [a,c) $ such that $f(d)=0$ ? ( I am kind of trying to go in reverse direction of Rolle's theorem ...)