Let $f(x,t)$ be a smooth function $\mathbb R^2\to\mathbb R$ such that $F_t(x):=f(x,t)$ has a unique local minimum in $x$ for every fixed $t\in[0,1]$. Further assume that this local minimum of $F_t(x)$ is also the unique global minimum of $F_t(x)$.
How regularly does the location of this unique minimum vary with respect to $t$? In other words, if $x=\chi(t)$ is the $x$-value where $F_t(x)$ attains its unique minimum, can we say that $\chi(t)$ is a smooth function of $t$? If not, is $\chi(t)$ differentiable or continuous?
I asked a similar version of this question here. The answer was correct and very clever, but I was wondering what happens if we insist that the unique global minimum was also a unique local minimum.