I don't have a clue with this problem. Thank you very much for your help & guidance.
(a) Suppose $F(x,t): X\times I \rightarrow R$ is a homotopy of Morse functions. That is, $f_t: X \rightarrow R$ is Morse for every $t$. Show that the set $C = \{(x,t)\in X\times I : d(f_t)_x = 0\}$ forms a closed, smooth submanifold of dimension one of $X\times I$. Assume the homotopy is constant near the ends of I and use an open interval.
(b) Let $\pi: X \times I \rightarrow I$. Show that $d(\pi|_C)_{(x,t)}: TC_{(x,t)} \rightarrow TI_t$ is surjective.
(c) Show that if $X$ is compact, there is no homotopy of Morse functions between two Morse functions with different numbers of critical points.
(a) Whenever you are asked to show that something defined by an equation is a smooth manifold, the Implicit Function theorem should come to mind. Let $n$ be the dimension of manifold $X$. Since being a submanifold is a local property, we can work with a coordinate patch: that is, consider $X$ as being $\mathbb R^n$. Then $d(f_t)$ is a smooth map into $\mathbb R^n$, let's denote it by $F:\mathbb R^n\times (0,1)\to \mathbb R^n$. We must prove that the derivative of $F$ has maximal rank at the points where $F=0$. This is where you will use the Morse condition.
(b) Between one-dimensional manifolds, "surjective derivative" is the same as "nonzero derivative". The implicit function theorem can tell you why it's nonzero.
(c) Use parts (a) and (b) to show that for every integer $k$, the set of values $t$ for which $f_t$ has exactly $k$ critical points, is open.