Critical points in a simple case of manifold with boundary.

Question

Let $X$ be a smooth, connected, compact manifold and $Y=X\times [a,b]$. Let $f:Y\to\mathbb{R}$ be a Morse function on $Y$ such that $f|_{X\times \{a\}}=a$ and $f|_{X\times \{b\}}=b$. What can we say about critical points of $f$? I'm looking for a version of Morse inequalities or Poincare-Hopf theorem siutable for this situation.