TITLE: A smooth function $f(x)$ has a unique local and global minimum. What happens to its location as $f(x)$ varies smoothly in time?
QUESTION [9 upvotes]: 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.

REPLY [4 votes]: For fixed $t$, $f(x,t) = (x^3−t)^2$ has its unique local and global minimum at $x = t^{1/3}$. This is not differentiable at $t=0$.