at.algebraic topology - Contractible manifold with boundary - is it a disc?

Given a function $psi:mathbb Rto mathbb R$,
set
$$Psi=psicircmathrm{dist}_ {partial M}, f=Psicdot(R-mathrm{dist}_ p)$$
for some fixed $R>mathrm{diam}\, M$.

Further,
$$d\,f=
(R-mathrm{dist}_ p)cdot d\,Psi-Psicdot d\,mathrm{dist}_ p$$
Thus, we may choose smooth increasing $psi$,
such that $psi(0)=0$
and it is constant outside of little nbhd of $0$ so that
$Psi$ is smooth.
(It is possible since the function $mathrm{dist}_ {partial M}$ is smooth and has no critical points in a small neighborhood of $partial M$.)
Note that $d\,Psi$ is positive muliple of $d\,mathrm{dist}_ {partial M}$.
Thus $d_x\,f=0$ means that geodesic from $x$ to $p$ goes directly in the direction of minimizing geodesic from $x$ to $partial M$, which can not happen.

Now we can apply Morse theory for $f$...